THE LEXINGTON 

GEAR WORKS, 

LEXINGTON, MASS. 

GEAR WHEELS AND GEAR CUTTING 
OF EVERY DESCRIPTION. 



List Brass Cm 

by mail al 

List Iron Gear 
List Iron Gear 
A Treatise on 



LIBRARY OF CONGRESS. 

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UNITED STATES OF AMERICA. 



ed in stock and sent 



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bles of data, etc. 



Job Gear Cutting ana jnaamng or every description and all sizes. 



SPUR GEARS, 
BEVEL GEARS, 
RACKS, 
RATCHETS, 
INTERNAL GEARS, 
SKEW BEVEL GEARS, 



ELASTIC GEARS, 
CHAIN GEARS, 
SPIRAL GEARS, 
ELLIPTIC GEARS, 
PINION WIRE, 
SHEET- BRASS GEARS, 



GEAR CUTTING MACHINERY. 



OUR GEAR CATALOGUE WILL BE SENT, AT REQUEST, TO 
ANYONE INTERESTED. 



GRANT'S 

Outline D0NT0GRA]p H. 

A collection of three hundred actual and accurate tooth outlines, to be cut 
from the paper and used as templets to guide the pencil. 

There are fifteen outlines in each 

pitch, all of the same size of tooth. 

and for the following numbers of teeth : 

10, 11, 12, 13-14, 15-16, 17-18, 19-21, 

22-24, 25-29, 30-36, 37-48, 49-72, 

73-144, 145-300, rack. 

The outlines are made in two 
shapes, the involute and the epicycloidal, 
and for the following pitches of each 
shape : 

Involute teeth for 1, 1\, 14, If, 2. 2|, 

24, 3. 34, or ^ diametral pitch. 
Epicycloidal teeth for 3, 2|, 24, 2|, 2, 
l|. 14, I4, 1, or % inch circular pitch. 
Published in one book, thick paper, 
containing four impressions of each of 
the three hundred outlines. Price, $2.00, 
post-paid. 

Also published in sheets, each sheet having the fifteen outlines of any one 
pitch. Price, 25 cents for any single sheet; any four for 50 cents. 




THE LEXINGTON GEAR WORKS, 

LEXINGTON, MASS., 

Gear Wheels and Gear Cutting of every Description. 

List Brass Gears. 184 sizes small brass ready-made gears. Sent free of 
postage by return mail. Prices very reasonable. 

List Iron Gears with Cast Teeth. 87 sizes iron cast gears. 

List Iron Gears -with Cut Teeth. 561 standard sizes iron cut gears. 
Many stock sizes finished and carried in stock. Any gear made to short order. 

Gear Cutting Machines. 

Treatise on Gear Wheels. 



JOB GEAR CUTTING AND MAKING. 

SEND FOR OUR CATALOGUE. 



2001! 



ODONTICS 



OR THE THEORY AND PRACTICE OF THE 



TEETH of GEARS, 

BY GEORGE B. GRANT. 





l^C/ 



PUBLISHED BY 

THE LEXINGTON GEAR WORKS, 

LEXINGTON, MASS. 
1891 

Co 



r,1 b . 



PRICK, $1.50, Post=paid. 



AGENTS WANTED. 

Write for Terms. 



Contributed to trie 

AMERICAN MACHINIST, 
For 1890. 



Copyright 1891, 
By GEORGE B. GRANT. 



A Working" Course of Study. 

It is not necessary that the student, especially if he is a workman, should 
learn all that is taught in this book, for it contains much that is not only difficult 
but also of minor practical importance. 

The beginner is therefore advised to master only the following sections : 

i, 2, 7 to 15, 22, 25, 31, 32 of the general theory; 

35 to 47 of the spur gear ; 

53 to 64 of the involute tooth ; 

76, 77, 80 to 83, 89 of the cycloidal tooth ; 

91, 95, 97 of the pin tooth ; 

98, 99, in, 113 to 119 of spiral and worm gears; 

154, 15), 158, 161 to 169 of the bevel gear. 

These include not half of the whole matter, but, knowing this much well, 
the student has a good outline knowledge of the whole, and he can then take 
the balance at leisure. 




ODONTICS 

OR THE THEORY AND PRACTICE OF THE 

TKKTH OF GKARS, 



lo THEORY OK TOOTH ACTION. 



1 . — INTRODUCTORY. 



The present object is practical, to reach 
and interest the man that makes the thing 
written of ; the machinist or the millwright 
that makes the gear wheel, or the drafts- 
man or foreman that directs the work, and 
to teach him not only how to make it, but 
what it is that he makes. 

To most mechanics a gear is a gear. 

" A yellow primrose by the shore, 
A yellow primrose was, to him, 
And it was nothing more ; " 

and, in fact, the gear is often a gear and 
nothing more, sometimes barely that. 

But, if the mechanic will look beyond the 
tips of his fingers, he will find that it can 



be something more ; that it is one of the 
most interesting objects in the field of scien- 
tific research, and not the simplest one ; that 
it has received the attention of many cele- 
brated mathematicians and engineers ; and 
that the study of its features will not only 
add to his practical knowledge, but also to 
his entertainment. There is an element in 
mathematics, and in its near relative, theoreti- 
cal mechanics, that possesses an educating 
and disciplining value beyond any capacity 
for earning present money. The thinking, 
inquisitive student of the day is the success- 
ful engineer or manufacturer of the future. 



2. — METHOD. 



The method will be fitted to the object, and 
will be as simple and direct as possible. It is 
not possible to treat all the items in simple 
every-day fashion, by plain graphical or arith- 
metical methods, but where there is a choice 
the path that is the plainest to the average 
intelligent and educated mechanic will be 
chosen. 

A thousand pages could be filled with the 
subject and not exhaust anything but the 
reader thereof, but what is written should 
receive and deserve attention, and must be 
condensed within such reasonable limits, that 
it shall not call for more time and labor than 



its limited application will warrant. Demon- 
strations and controversies will be avoided, 
and the matter will be confined as far as is 
possible to plain statements of facts, with 
illustrations. The simplest diagram is often 
a better teacher than a page of description. 
First, we shall study the odontoid or pure 
tooth curve as applied to spur gears, then 
we shall consider the involute, cycloid, and 
pin tooth, special forms in which it is found 
in practice ; then the modifications of the 
spur gear, known as the spiral gear, and the 
elliptic gear ; then the bevel gear, and lastly 
the skew bevel gear. 



Literature. 



3. — PARTICULARLY IMPORTANT. 



Begin at the beginning. 

The natural tendency is too often to skip 
first principles, and begin with more ad- 
vanced and interesting matter, and the result 
is a trashy, knowledge that stands on no 
foundation and is soon lost. When a fact 
is learned by rote it may be remembered, 
but when it follows naturally upon some 
simple principle it cannot be forgotten. 

Therefore the student is urged to begin 
with and pay close attention to the odontoid 
or pure tooth curve, before going on to its 



special applications, for the apparently dry 
and trivial matter relating to it is really the 
foundation of the whole subject. 

The usual course is to begin at once with 
the cycloidal tooth, to hurry over the in- 
volute tooth, and then, if there is room, it 
is stated that such curves are particular 
forms of some confused and indefinite general 
curve. Our course will be to study the unde- 
fined tooth curve first, and then take up 
its particular cases. 



4. — LITERATURE. 



It is impracticable to acknowledge all the 
sources from which information has been 
drawn, but it is in order to briefly mention 
the principal works devoted to the subject. 

Professor Herrmann's section of Professor 
Weisbach's ' ' Mechanics of Engineering and 
Machinery" is the most important work that 
can be named in this connection. It treats 
of much besides the teeth of gears, but its 
treatment of that branch is particularly 
valuable. It is not easy reading. Wiley, 
$5.00. 

Professor Willis' "Principles of Mechan- 
ism " is a celebrated book, now many years 
behind the age, but it is, nevertheless, of the 
greatest value and interest in this matter. To 
Willis we are indebted for many of the most 
important additions to our knowledge of 
theoretical and practical mechanism. Long- 
mans, $7.50. Out of print. 

Professor Rankine's " Machinery and Mill- 
work " should not be neglected by the 
student, for, although it is the dryest of 
books, its value is as great as its reputation. 
Griffin, $5.00. 

Professor MacCord's "Kinematics" is a 
work that abounds in novelties, and is writ- 
ten in an attractive style. It contains many 
errors, and some hobbies, and needs a thorough 
revision, but the student cannot afford to 
avoid it, or even to slight it. Wiley, $5.00. 

Mr. Beale's ' ' Practical Treatise on Gear- 
ing " is really practical. Many of the so-call- 
ed "practical" books are neither practical or 
theoretical, but we have in this small book 
a collection of workable information that 



should be within the reach of every man 
who pretends to be a machinist. We have 
drawn from it, by permission, particularly 
with regard to spiral and worm gears. 
Mr. Beale's experimental work, in connection 
with the spiral gear, has been of great 
service. The Brown & Sharpe Mfg. Co., 
$2.10. 

Professor Reuleaux's ' ' Konstrukteur " is a 
justly celebrated work in the German lan- 
guage. A translation of it is now being 
published in an American periodical — Me- 
chanics. 

Professor Klein, the translator of Herr- 
mann's work, has lately published the " Ele 
ments of Machine Design," a collection of 
practical examples, with illustrations. J. F. 
Klein, Bethlehem, Pa., $6.00. 

"Mill Gearing," by Thomas Box, is a 
practical work by an engineer, and from it 
we have drawn much of our matter on the 
cloudy subject of the strength and horse- 
power of gearing. Spon, $3.00. 

"Elementary Mechanism," by Professors 
Stahl and Woods, is a recent work of general 
merit. It is well designed as a text book, 
and treats the subject in a simple and in- 
teresting manner. Van Nostrand, $2.00. 

In addition to the above works, reference 
may be made to numerous articles to be 
found in periodicals, notably in the ' ' Ameri- 
can Machinist," the " Scientific American 
Supplement," the "Journal of the Franklin 
Institute," " Mechanics," and the "Transac- 
tions of the American Society of Mechanical 
Engineers." 



General Theory . 



5. — KINEMATICS. 



This, the science of pure mechanism, re- 
lates exclusively to the constrained and 
geometric motions of mechanism, and it 
has nothing to do with questions of force, 
weight, velocity, temperature, elasticity, 
etc. The path of a cannon ball is not with- 
in the field of kinematics, because it depends 
upon time and force. A belt and pulley are 
kinematic agents, because the contact be- 
tween them can be assumed to be definite, 



and the action is therefore geometric, but 
the slipping and stretching of the belt is not 
kinematic. The action of gear teeth upon 
each other is purely kinematic, but we can- 
not consider whether the material is wood, or 
steel, or wax, whether the gears are lifting 
one pound or a ton, or whether they are run- 
ning at one revolution per second or one per 
day. 



6. — ODOTSTTICS 

The name ' ' odontics ' ; may be selected 
for that limited but important branch of 
kinematics that is concerned with the trans- 
mission of continuous motion from one 
body to another by means of projecting 
teeth. 

Even this restricted corner of the whole 
subject is too large for the present purpose, 
for it covers much that cannot be considered 
within our set limits, and gear wheels must, 
therefore, be defined as devices for trans- 
mitting continuous motion from one fixed 
axis to another by means of engaging teeth. 

Thus confined, gear wheels may be con- 
veniently divided into three general classes. 

Skew bevel gears, transmitting motion be- 
tween axes not in the same plane. 

Bevel gears, transmit ling motion between 
intersecting axes. 



Spur gears, transmitting motion between 
parallel axes. 

The last two classes are particular cases of 
the first; for, if the shafts may be askew at 
any distance, that distance may be zero, and 
if they intersect at any point, that point may 
be at infinity. 

It would be scientifically more correct to 
first develop the skew bevel gear, and from 
that proceed to the bevel and spur gear, but 
practical clearness and convenience is often 
more to be admired than strict accuracy, 
and, as the true path is difficult to follow, we 
shall enter in the rear, and consider the spur 
gear first. 

Odontics does not properly include the 
consideration of questions of strength, pow- 
er and friction, but we must admit certain 
important items in that direction. 



7. — PITCH 

The fixed axes are connected with each 
other by imaginary surfaces called "axoids," 
or pitch surfaces, touching each other along 
a single straight line. We must imagine 
that the pitch surfaces roll on each other 
without slipping, as if adhering by friction. 

The whole object of odontics is to provide 
these imaginary surfaces with teeth, by 



SURFACES. 

which they can take advantage of the 
strength of their material and transmit 
power that is as definite as the geometric 
motion. 

The pitch surface of the skew bevel gear 
is the hyperboloid of revolution, which be- 
comes a cone when the axes intersect, and a 
cylinder when the axes are parallel. 



8. — NORMAL SURFACES. 



An important adjunct of the pitch surface 
is the normal surface, or surface that is 
everywhere at right angles to both pitch sur- 
faces of a pair of axes, and upon which the 
action of the teeth on each other may best 
be studied. 



For the skew bevel gear there does not 
appear to be any normal surface. For the 
bevel gear the normal surface is a sphere, 
and for the spur gear the sphere becomes a 
plane. 



General Theory. 



9. — UNCERTAINTIES. 



The theory of tooth action is not yet full 
and definite in all its parts, for there are 
some disputed points, and some confusion 
and clashing of rules and systems. This is 



particularly the case with the theory of spiral 
and skew bevel teeth, for much of the work 
that has been done is clearly wrong, and 
there is little that has been definitely decided. 



10. — PITCH 



Fig. 1. 




Pitch cylinder* 

Two cylinders, A and B, Fig. 1, that will 
roll on each other, will transmit rotary mo- 
tion from one of the fixed parallel axes c and 
C to the other, if their surfaces are provided 
with engaging projections. 

When these projections are so small that 
they are imperceptible, the motion is said to 
be transmitted by friction, and it is prac- 
tically uniform. But when they are of 
large size, and readily observed, the motion, 



CYLINDERS. 

although it is unchanged in nature, is said to 
be transmitted by direct pressure, and it is 
irregular unless the acting surfaces of the 
projections are carefully shaped to produce 
an even motion. 

The whole object of odontics is to so shape 
these large projections or teeth that they 
shall transmit the same uniform motion be- 
tween the rotating cylinders, as would be 
apparently transmitted by friction. 

These cylinders are imaginary in actual 
practice, although they are one of the 
principal elements of the theory, and they 
are called the axoids, or pitch cylinders of 
the gears. 

The normal surface (8) of the spur gear is 
a plane, and, as all sections by normal sur 
faces are alike, we can study the action on 
a plane figure easier than in the solid body 
of the gear. 



Fig. 2. 



11. — THE LAW OF TOOTH CONTACT. 

The common normal to the tooth curves must 
pass through the pitch point. 

That is, in Fig. 2, if the tooth curves OD 
and o d are to transmit the same motion 
between the pitch lines pi and PL as 
would be transmitted by frictional contact 
at the pitch point 0, they must be so shaped 
that their common normal Op at their com- 
mon point p shall pass through that pitch 
point. 

Conversely, if the tooth curves are so 
shaped that their common normal always 
passes through the pitch point, they will 
transmit the required uniform motion. 




Tooth action 



With the above conditions given we can 
deduce the following la,w: 



12. — THE ODONTOID. 



This universal law enables us to define the 
"odontoid," or pure tooth curve, for the 
contact of the pitch lines at the pitch point 
is continuous and progressive, and, if the 
tooth curves are to transmit the same motion, 
their normals must be arranged in a contin- 



uous and progressive manner. The normals 
nl, as in Fig. 3, must be arranged without 
a break or a crossing, not only springing 
from the odontoid at consecutive points, but 
intersecting the pitch line at consecutive 
points. This arrangement may be called 



General Theory. 




Fig. 3. 

Odontoid* 

"consecutive," and the definition is not a 
law by itself, but an expression of the given 
universal law. 

It is seen that the odontoid is inseparably 
connected with its pitch line, and that the 
same curve may be an odontoid with re : 
spect to one pitch line, and not with respect 
to some other. The curve Fig. 4 is an 
odontoid with respect to the pitch Muepl, 




but not with respect to the pitch line pi' be- 
yond the point p at which the normal is tan- 
gent to that pitch line. 

The odontoid, so far as defined, is not a 
definite thing, and, for practical purposes, it 
must be given some particular shape. It 
may be involute or cycloidal, or of other 
form, but must always have normals ar- 
ranged in consecutive order. 



13. — THE LINE OF ACTION 

As the tooth curves od and OB, Fig. 5, 
work together, the point of contact will 
travel along a line Op W called the "line of 
action." 

There is a definite relation between the 
odontoid and the line of action, so that, if 
either one is given, the other is fixed. If 
the odontoid OD is given, with its pitch 
line PL, the line of action is determined 
without reference to the pitch line pi or its 
odontoid; and, conversely, if the pitch line 
and line of action are given, the odontoid to 
correspond is determined. 




Line of 
action 



14. — INTERCHANGEABLE ODONTOIDS. 



This feature leads at once to the broad 
and useful fact that all odontoids, on pitch 
lines of all sizes, that are formed from the 
same line of action, will work together inter- 
changeably, any one working with any other. 



Therefore, to produce an interchangeable 
set of odontoids we can choose any one line 
of action, and form any desired number of 
them from it. 



15. — INTERNAL CONTACT. 



The pitch lines of Fig. 5 curve in opposite 
directions, and the contact is said to be " ex- 
ternal." But the principles involved are in- 
dependent of the direction of the pitch lines, 
and they may curve in the same direction, as 
in Fig. 6, in "internal" contact. 

Tooth contact is between lines only, there 
being no theoretical need of a solid material 
on either side of the line, so that either side 




Internal action 



6 



Cusps and Terminals. 



of the tooth may be chosen as the practical 
working side. 

Therefore the internal gear is precisely like 
the external gear of the same pitch diameter, 
working on the same lines of action, so far 
as the odontoids are concerned, as illustrated 
by Fig. 7. 



Fig. 7. 




Internal and 
external teeth 



16. — THE CUSP AND INTERFERENCE. 

When, as in Fig. 8, the pitch circle p I is 
so small with respect to the line of action 
C G" W, that two tangent circles C c' and 
C" c" can be drawn to the line of action from 
the center G of the pitch line, we shall have 
a troublesome convolution in the resulting 
flank curve o d. This convolution will be 
formed of two cusps, a first cusp c' on the 
inner tangent arc, the "base circle" G' c' , 
and a second cusp c" on the outer tangent 
arc G" c". 

This happens with any form of odontoid, 
although sometimes in disguised form, and 
creates a practical difficulty that can be 
avoided only by stopping the tooth curve at 
the first cusp d . 

Furthermore, any odontoid OD that is to 
work with the odontoid o d, must be cut off 
at the point k on the "limit line" G' Jc 
through the point G' from the center c. 

If the odontoids, when the pitch line is so 
small that the cusps occur, are not cut off as 




required, the action will still be mathemati- 
cally perfect, but, as the contact changes at a 
cusp, from one side of the curve to the other, 
the action is no longer practicable with solid 
teeth. The curves will cross each other, 
and there will be an interference. 



17. — THE SMALLEST PITCH CIRCLE. 



To determine the smallest pitch circle that 
can be used, and avoid the cusps altogether, 
find by trial the point G, Fig. 9, from which 
but one tangent arc G c' can be drawn to the 
line of action C' W. This point will be 
the center of the smallest pitch circle, and 
all points outside of it will avoid interference, 
while all inside of it will be subject to it. 




Smallest 
radius 



18. — THE TERMINAL POINT 

When a tangent arc can be drawn, from 
the pitch point as a center, to the line of 
action at any point T, except the vertex W, 
Fig. 10, there will be a corresponding cross- 
ing of the normals to the odontoid commenc- 
ing at the poif't t, and a termination of the 
action when the point t reaches the point T. 

As the action approaches the terminal 
point I 1 there will be two points of aciion, 




Fig. 10. 



Terminal point 



Secondary Action. 



since the odontoid crosses the line of action 
at two points — one point of direct and ordi- 
nary action at 8, and another point of retro- 
grade and unusual action at V. These two 
points of action will come together at T, the 
odontoid will leave the line of action, and all 



tooth action will then cease. The retro- 
grade action is theoretically and actually 
correct, but it is so oblique that it is of 
no practical value, and therefore the odon- 
toid may as well be cut off at its terminal 
point t. 



19. — SPEED OF THE 

Lay off 8, Fig. 5, to represent the speed 
of the pitch lines, and draw 8 A at right 
angles with the common normal p. Draw 
p G tangent to the line of action at the point 
of action p. 

Lay off p B equal, to A, and draw B G 
at right angles to B. Then p G will be 
the speed of the point of action along the 
line of action. 



POINT OP ACTION. 

When the line of action is a circle the 
angle 8 A is always equal to the angle 
B p G, and therefore the speed of the point 
of action is uniform, and equal to that of the 
pitch lines. 

If the line of action is a straight line the 
angle Bp Cwill be constant — always zero— 
and therefore the speed of the point of action 
will be uniform and always equal to A. 



20. — NATURE OP THE TOOTH ACTION. 



The nature of the action may be deter- 
mined by a study of the normal intersections; 
the intersections of the normals with the 
odontoid being at uniform distances apart, 
their intersections with the pitch lines will 
indicate the action of the teeth. If the nor- 



mal intersections, as in Fig. 3, are quite regu- 
lar, the action of the teeth will be smooth 
and regular, while if they are crowded with- 
in a narrow space the action of the tooth will 
be crowded and jerky. 



21. — THE SECONDARY LINE OP ACTION. 



From the universal law of tooth contact 
stated in (11) we can reason that any 
point on the tooth curve is in position for 
contact whenever its normal passes through 
the pitch point 0, and therefore that the 
point will then be upon a line of action. 

In Fig. 11 the normal to the point p must 
cross the pitch line twice— at a primary in- 
tersection a, and at a secondary intersection 
b, and therefore there will be a point of 
action on a primary line of action M 1 at q, 
when the curve has moved so that the pri- 
mary point of intersection a is at the pitch 
point 0, and a point of action wona second- 
ary line of action, when the secondary point 
of intersection b has reached the pitch point. 

Therefore there will generally be not only 
the primary line of action q M or q' M' , 
but also a secondary line w Tor w' Y'. 

The secondary line of action must have the 
same property as the first, as a locus of con- 
tact, and therefore if we can so arrange two 
pitch lines with their odontoids that their 
secondary lines of action coincide, there will 
be secondary contact between the odontoids. 




Secondary 
actio' > 



When it so happens that both primary and 
secondary lines coincide, we shall have 
double contact. Two points of contact will 
exist at the same time, one on the primary 
and the other on the secondary line of action. 

The secondary lines of action cannot be 
made to coincide when the contact is exter- 
nal, but when it is internal they sometimes 
can be, so that the matter has an application 
to internal gears. 



8 



Interchangeable Tooth. 



It is to be noticed that the primary line is 
independent of the pitch line, while the sec- 
ondary is dependent upon it. 



Secondary contact is an interesting feature 
of tooth action, but it is of small importance, 
and has been studied but little. 



22. — THE INTERCHANGEABLE TOOTH. 



The simple odontoid so far studied is the 
perfect solution of the problem from a 
mathematical point of view, for it will trans- 
mit the required uniform motion as long as 
it remains in working contact. But from a 
mechanical point of view it is still incom- 
plete, as it works in but one direction, 
through but a limited distance, and, although 
the odontoids are interchangeable, the gears 
are not. 

In order that the gears shall be fully in- 
terchangeable, it is necessary that the teeth 
shall have both faces and flanks, and that the 
line of action for the face shall be equal to 
that for the flank; that is, the tooth must 
have an odontoid on each side of the pitch 
line, the face o d, Fig. 12, outside, and the 
flank o d' inside of it, and the line of action 
I a for the faces must be like the line of 
action I a' for the flanks. If so made, any 
gear will work with any other, without re- 
gard to the diameters of the pitch lines. 

But such a gear will run in but one direc- 
tion, and to make it double-acting it must 
have odontoids facing both ways, as in Fig. 
ID. Gears so made will be both double-act- 
ing and interchangeable, and it is not neces- 
sary that both sides of the tooth shall be 
alike. 

Again, the unsymmetrical gear of Fig. 13 
fails when it is turned over, upside down , 
for then the unlike odontoids come together, 
and, to avoid this last difficulty, all four of 
the lines of action must be alike, producing 
the complete and practically perfect tooth of 
Fig. 14. 

We can therefore define the completely in- 
terchangeable tooth, as the tooth that is 
formed from four like lines of action. 



Fig. 12. 




Interchangeable 
teeth 




Fig. 13. 



Unsymmetrical teeth 

Fig.14. 




Complete teeth 



. — INTERCHANGEABLE RACK TOOTH. 



When the pitch line is a circle the flanks of tween face and flank. We then have the im- 



the tooth are not like the faces, but when it 
is a straight line there is no distinction be- 



portant practical fact that the four odontoids 
of the interchangeable rack tooth are alike. 



Construction by Points. 



24. — CONSTRUCTION BY POINTS. 

When we have an odontoid and its pitch 
line given, it is a very simple matter to con- 
struct either the line of action or the conju- 
gate odontoid for any other pitch line. 

We know, for example, the odontoid s p, 
Fig. 15, on the pitch line p I, and it is re- 
quired to construct an odontoid on the pitch 
line P L that is conjugate to it. 

As the odontoid is given we know or can 
construct its normals. Construct the normal 
p a from any chosen point p, draw the radial 
line da C, lay off A equal to a 0, draw the 
radial line A G, lay* off the anetfe NAD 
equal to the angle n a d, lay off Pi equal 
to p a, and P will be a point in the required 
conjugate odontoid 8 P. P A will be a 
normal to the curve. Construct a number of 
points by this process, and draw the required 
curve through them. The tangents * t and 
8 T make equal angles with the pitch lines, 
so that the required curve can often be fully 
determined by drawing its tangent and one 
or two points. 

To construct the line of action, make the 
angle m e equal to the angle n a d, 
and lay off q equal to p a. The point 
q is on a circle from either p or P drawn 
from the centers C, and is the point at 
which p and P will coincide when the two 
curves are in working contact, the normals 
p a and P A then coinciding with the 
radiant q. 




Construction! 
by points 



When the line of action alone is given, the 
odontoids for given pitch lines are fully de- 
termined, but there seems to be no simple 
graphical method for constructing them ex- 
cept for special cases. They can be obtained 
by the use of the calculus (33), or drawn by 
the integrating instrument of (34). 

The two tooth curves thus constructed are 
paired, and are said to be '•'conjugate" to 
each other. 



25. — THE ARC 
The action between two teeth commences 
and ends at the intersections m and N of the 
line of action with the addendum lines of the 
two gears, a I and A L, Fig. 16. The arc 
of action is the distance ab on the pitch line 
that is passed over by the tooth while it is in 
action. 

The arc a passed over while the point of 
contact is approaching the pitch point, is 
called the arc of approach, and b, that 
passed over while the action is receding from 
that point, is the arc of recess. 

With a given line of action the arcs of ap- 
proach and recess can be controlled by the 
addenda. If it is desirable to have a great 
recess and a small approach, the addendum 



OP ACTION. 




of the gear that acts as a driver is to be in- 
creased. When there is a limit line (16), it 
limits the addendum and the arc of action. 



10 



Molding Processes. 



26.— OBLIQUITY 

When a pair of teeth bear upon each 
other, the direction of the force exerted "be- 
tween them is that of the common normal 
Op, Fig. 17, and passes through the pitch 
point 0. Except when the point of contact 
is at the pitch point the direction of the 
pressure will deviate from the normal to the 
line of centers by the angle of obliquity 
Z p, and with many forms of teeth the 
angle is never zero. 

The force exerted between two teeth at 
their point of contact is found by laying off 
the tangential force H with which the driv- 
ing gear D is turning, and drawing the line 
H V parallel to the line of centers, to find 
the force V — P K. It is proportional to 
the secant of the angle of obliquity, and in- 
creases rapidly with that angle. 

The chief influence of the obliquity is 
upon the friction between the teeth, and con- 
sequent inefficiency of the gear, and upon 
the destruction by wearing. It is par- 
ticularly important upon the approaching 
action, and a gear that is otherwise perfect 
may be inoperative on account of excessive 
obliquity. 

Although the direct pressure of the teeth 
upon each other at their point of contact 



OF THE ACTION 
K 




^^ Fig. 17. 

will vary with the obliquity, the tangential 
force exerted to turn the gear is always 
uniform. Leaving friction out of the calcu- 
lation, the two gears of a pair always turn 
with the same force at their pitch lines. 

The obliquity of the action has an effect 
upon the direction and amount of the 
pressure of the gear upon its shaft bearing, 
but the usual variation is of little conse- 
quence. 

It is desirable that the pressure between 
the teeth should be as uniform as possible, 
not only in amount, but in direction, and 
excessive obliquity is to be carefully avoided. 



27.— CONSTRUCTION BY MOLDING. 



The mode of action of the conjugate teeth 
upon each other, suggests a process by which 
a given tooth can be made to form its conju- 
gate by the process of molding. 

The given tooth, all of its normal sections 
being of some odontoidal form, is made 
of some hard substance, while the blank in 
which the conjugate teeth are to be formed 
is made of some plastic material. The shafts 
of the two wheels are given, by any means, 
the same motions as if their pitch surfaces 
were rolled together. The hard tooth will 
then mold the soft tooth into the true conju- 
gate shape. 



It matters not what shape is given the 
molding tooth, if its sections are all odon- 
toidal, and a twisted or irregular shape will 
be as serviceable as the common straight tooth. 

This process is continually in operation be- 
tween a pair of newly cut teeth, or between 
rough cast teeth, until the badly matched 
surfaces have been worn to a better fit, but 
it is too slow for ordinary purposes, and is 
of little practical value. 

Gears can be formed by this process, by 
rolling a steel forming gear against a white 
hot blank, but the process can hardly be 
called practical. 



28. — MOLDING PLANING PROCESS. 



Although the described molding process 
is of limited practical value, having but one 
direct application, it leads to a process of 
great value when the tooth is straight or of 



such a shape that it can be followed by a 

planing tool, its normal sections being alike. 

The originating tooth is fixed in the shape 

of a steel cutting tool C, Fig. 18, which is 



Pla?iing Processes. 



11 



rapidly reciprocated in guides G, in 
the direction of the leugih of the 
100th, as the two pitch wheels A and. 
B are rolled together. Although the 
tool has bui a single cutting edge, its 
motion mak^s it the equivalent of the 
molding 100th, and it will plane out 
the conjugate tooth D by a process 
that is the equivalent of the more 
general molding process. 

A simple graphical method is 
founded upon this molding process, 
the shaping tool taking the form of a 
thin template C, Fig. 19, that is re- 
peatedly scribed about as the pitch 
wheels are rolled together, the marks 
combining to form the conjugate 
tooth curves D. 

This mechanical process has the 
decided advantage over the process 
of construction by points (24), that 
the tooth is formed with a correct 
fillet (44), and is much stronger. 
The dotted lines show the tooth that 
would be constructed by points. 

The only practicable method for 
forming the line of action when this 
method is used is by observing and 
marking a number of points of con- 
tact between the teeth. This method 
is applicable to all possible forms of 
spur teeth, either straight, twisted or 
spiral. It can be practically applitd 
only to the octoid form of bevel tooth. 

On account of the fillet (44) that 
is formed by this process, the tooth 
space cannot be used with a mating 
gear having more teeth than that of 
the forming gear, although it belongs 
to the same interchangeable set. The 
tooth space of the figure will not run 
with a tooth on a pitch line larger 
than the pitch line A. 

Therefore the rack tooth must be 
useu as the forming tooth, to allow of 
the use of all gears of the set up to 
the rack. Gears of the set thus formed 
will not work with internal gtars. 



Hlolding planing 
tnctltod 




Fig. 18 t 




Graphical molding 
method 



29. — LINEAR PLANING PROCESS. 



A second planing process, quite 
distinct from the molding process of 
(27), is founded upon the fact that 
the tooth curves are in contact at a 
single point which has a progressive 
motion along the line of action. 

Therefore if a single cutting point 
p, Fig. 20, is caused to travel along 
the line of action with the proper 
speed relatively to the speed of the 
pitch line, it will trim the tooth out- 
line to the proper odontoidal shape. 

The figure shows the application to 




Fig. 20* 



Linear planing 
method 



12 



Particular Forms. 



the involute tooth, the path of the cutting 
point being the straight line I a, and its speed 
being the speed of the base line b I. 

When the cutting point follows the circu- 
lar line of action with a speed equal to that 
of the pitch line, it will plane out the 
cycloidal tooth curve. 



This process is applicable to all possible 
forms of gear teeth, either spur or bevel, in 
either external or internal contact. 

When the curvature of the odontoid will 
permit, the milling cutter may take the place 
of the planing tool, and is the equivalent of it. 



30. — THE BACK ORIGINATOR. 



The molding planing process of (28) sup- 
plies a means for easily and accurately pro 
ducing an interchangeable set of gears or 
cutters for gears, and it is best applied by 
means of the rack tooth as the originator. 
All four curves of the rack tooth being alike, 
the tooth is easily formed, particularly for 
the involute or the segmental systems, and it 
is a matter of less consequence that the curves 



shall be of some particular form, if care is 
taken that it is odontoidal. 

It has been taught, and it is therefore some- 
times considered, that any " four similar and 
equal lines in alternate reversion" will an- 
swer the purpose, but it is necessary that the 
four similar curves shall be odontoids. Four 
circular arcs, with centers on the pitch line, will 
answer the definition, but are not odontoids. 



31. — PARTICULAR FORMS OF THE ODONTOTD. 



The odontoid, as so far examined, is un- 
defined except as to one feature of the ar- 
rangement of its normals, and to bring it 
into practical use it is necessary to give it 
some definite shape. This is most easily ac- 
complished by choosing some simple curve 
for the rack odontoid, and from that making 
an interchangeable set. A more correct but 
much more difficult method would be to 
choose some definite line of action, and from 
that derive the odontoids. 

If the rack odontoids are straight lines, 
Fig. 21, the common involute tooth system 
will be produced, and the line of action will 
be a straight line at right angles with the 
rack odontoid. For bevel teeth, as will be 
shown, the straight line odontoid produces 
the octoid tooth system, while to produce the 
involute system it is necessary to define the 
line of action as a straight line, and derive 
the system from thai. 

If the rack odontoids are cycloids, as in 
Fig. 22, the resulting tooth system will be 
the cycloidal, commonly misnamed the 
" epicycloidal " system. The line of action 
will be a circle equal to the roller of the 
cycloid. 

If the rack odontoids are segments of cir- 
cles from centers not on the pitch line, but 
inside of it, as in Fig. 23, the tooth system 



Fig. 21. 




Fig. 



Rolled Curve Theory. 



13 



will be the segmental, and its line of action 
will be the loop of the " Conchoid of Nico- 
medes." 

If we choose a parabola for the rack tooth, 
as in Fig. 24, the parabolic sysiem will be 
formed with its peculiar ''hour glass" line 
of action. 

Only three of these tooth systems are in 
actual use, the involute and the cycloidal for 
spur gears, and the octoid for bevel gears 
only, and we will therefore confine the ap- 
plication of the theory to them. 

Only one of the systems in common use 
for spur gears, the involute, should be in use 
at all, and we will pay principal attention to 
that. 




Parabolic 

Fig* 24. 

The segmental system would be superior 
to the cycloidal, and in many cases to the in- 
volute; but as there is already one system too 
many, we will not attempt to add another. 



32. — THE ROLLED 

If any curve M, Fig. 25, is rolled on any 
pitch curve p I, a point p in the former will 
trace out on the plane of the latter a curve 
s p z, called a rolled curve. 

The line p q, from the tracing point p to 
the point of contact q, is a normal to the 
curve s p z, and, as all the normals are ar- 
ranged in "consecutive" order, that curve 
must be an odontoid. The converse of this 
statement is also true, that all odontoids are 
rolled curves ; but the fact is genera) 1 y \ ery 
far fetched and of no practical importance. 

It is al«o a property of all such curves 
that are rolled on different pitch lines, that 
they are interchangeable. 

Tnis accidental and occasionally useful 
feature of the rolled curve has generally 
been made to serve as a basis for the general 
theory of the gear tooth curve, and it is re- 
sponsible for the usually clumsy and limited 
treatment of that theory. The general law 
is simple enough to define, but it is so diffi- 
cult to apply, that but one tooth curve, the 
cycloidal, which happens to have the circle 
for a roller, can be intelligently handled 
by it, and the natural result is, that that 
curve has received the bulk of the atten- 
tion. 

For example, the simplest and best of 



CURVE THEORY. 




Moiled curve 

Fig. 25. 

all the odontoids, the involute, is entirely 
beyond its reach, because its roller is the 
logarithmic spiral, a transcendental curve 
that can be reached only by the higher mathe- 
matics. 

No tooth curve, which, like the involute, 
crosses the pitch line at any angle but a 
right angle, can be traced by a point in a 
simple curve. The tracing point must be 
the pole of a spiral, and therefore the trac- 
ing of such a curve is a mechanical impossi- 
bility. A practicable rolled odontoid must 
cross the pitch line at right angles. 

To use the rolled curve theory as a base of 
operations will confine the discussion to the 
cycloidal tooth, for the involute can only be 
reached by abandoning its true logarithmic 
roller, and taking advantage of one of its 
peculiar properties, and the segmental, 
sinusoidal, parabolic, and pin tooth, as well 
as most other available odontoids, cannot be 
discussed at all. 



33. — MATHEMATICAL RELATION OF ODONTOID AND LINE OP ACTION. 

In Fig. 26 the odontoid on the pitch line j by the relations P T = p t = y, and T S = 
p I is connected wilh the line of action I a, \t = x, where P 8 is the normal to the 



14 



MatJicjnatical Relation. 



odontoid at the point P, T £isa tangent to 
the pitch line at the intersection of the nor- 
mal, and P T is a normal to the tangent. 

When any odontoid is given by its equa- 
tion, that of the line of action can be found 
by a process of differentiation, and when 
the line of action is given by its equation, 
that of the odontoid can be found by a 
process of integration. 

These processes, for the general case where 
the pitch line is curved, are quite intricate, 
but when the pitch line is a straight line, 
they are simple, and may be worked as 
follows. 

To get the equation of the line of action 
from that of the given rack odontoid, ar- 
range the equation of the odontoid in the 
form x = f(y), and put its differential co- 

d x 11 

efficient - — equal to — . Thus, the equation 
ci y x 

of the straight rack odontoid of the involute 

system is y = x tan. A, from which 

d x 1 

d y tao. A 



= — , and y = - -r 

x a tan. A 



is 



the equation of the straight line of action at 
right angles to the odontoid. Again, the 
equation of the cycloid being x — ver. sin. -1 

y — \/2ry — y 2 , we have 

d x _ y _ _y_ 

dy \/ 2ry — y 2 x ' 




and x 2 -\-y 2 = 2ry is the equation of the 
circular line of action. 

To get the equation of the odontoid when 
that of the line of action is given, arrange 
the equation of the line of action in the form 



and inte- 



-f- =f(y)> Put it equal to mi- 
grate. Thus, the equation of the straight 
line of action being 

x 
y " lan. A ' 
we have 

y 1 dx 

x tan. A ~ dy' 

and y — x tan. 



tan. A 

A is the equation of the 
straight odontoid at right angles to the line 
of action. Again, the equation of the circu- 
lar line of action being x 2 -f- y 2 = 2ry, we 

have 

y_ _ y _d_x 

x *j2ry —y 2 d y' 

and x = ver. sin. -1 ; y — \^2ry 

cycloidal odontoid. 



y is the 



34. — THE ODONTOIDAL INTEGRATED,. 



The form of the odontoid to correspond to 
a given line of action and a given pitch line 
can be determined only by the integral cal- 
culus (33), it evidently being impossible to 
contrive a general graphical or algebraic 
method. 

But it can be directly drawn by an instru- 
ment, the principle of which is analogous to 
that of the well-known polar planimeter for 
integrating surfaces. 

The bar R, Fig. 27, moves at right angles 
to the line of centers, and it moves the pitch 
wheel A, with the same speed at the pitch 
line. The bar G has a point p, that is 
confined to move in the given line of action 
p W, and it is so guided that it always 
passes through the pitch point 0. 

The two bars bear upon each other by 
friction, and we must suppose that there 
is no other friction to oppose the motion of 
the bar C. 



Fig. 27. 




Odontoidal Integrator 



Then the pointy will trace out the odontoid 
spzupon the pitch wheel A, or upon any 
other pitch wheel B rolling with the bar B 
on either side of it. 



2. THE; SPUR. QEAR IN GENERAL. 

35. — THE CIRCULAR PITCH. 

1 4, covered by each gear of one-inch pitch to match others in 
use, and to be fourteen and a half inches 
diameter. The circumference of the pitch 
line is 45.53 inches, and he has his choice be- 
tween 45 and 46 teeth, both wrong. Per- 
haps the most frequent cause of error is 
that the workman is apt to apply a rule 
directly to the teeth of a gear he is about to 
repair or match, to get the circular pitch, 
and the result is more likely to be wrong than 
right. 

The best plan when using this unit is to 
get convenient pitch diameters and let the 
pitch come as it will, provided that gears 
that work together are of the same pitch, and 
that is simply a roundabout way of using the 
diametral pitch unit. 

When the circular pitch must be used the 
following fcible will greatly assist the work 
and save calculation. For example, the 
pitch diameter of a gear of three-quarter-inch 
pitch and 37 teeth is three-quarters the tabu- 
lar number 11.78, or 8.84 inches. 



The distance a 0, Fig 
tooth upon the pitch circle, is commonly 
called the "circular pitch," and often the 
"circumferential pitch." The term "pitch 
arc" is the most appropriate but is not in 
common use. 

This was formerly the measurement by 
which the size of the tooth was always 
stated, a tooth being said to be of a certain 
"pitch," and all of its other dimensions 
being expressed in terms of that unit, but it 
is fast being replaced, and should be entirely 
replaced, by the more convenient "diametral 
pitch " unit. 

The circumference of a circle is measured 
in terms of its diameter by means of an in- 
commensurable fractional number 3.14159, 
called n- (pi), and, therefore, if the tooth is 
measured upon the arc of the circle by means 
of the circular pitch, one of two inconveni- 
ences must be tolerated. Either the pitch 
must be an inconvenient fraction, or else the 
pitch diameter must be as inconvenient, for 
the gear cannot have a fractional number of 
teeth. The fractional calculations are so 
clumsy that a table of pitch diameters cor- 
responding to given numbers of teeth should 
be used, and errors in the laying out of the 
work are of constant occurrence. 

Again, outside of the liability of error in 
making calculations, the circular pitch sys- 
tem is a constant source of error in the hands 
of lazy or incompetent draftsmen or work- 
men, for there is a constant temptation, 
often yielded to, to foree the clumsy figures 
a little to produce some desired result. For 
example, a millwright has to make a gear of 
fourteen inches pitch diameter with fourteen 
teeth. He finds by the usual computation 
that the circular pitch is 3.14 inches, and, as 
his odontograph has a table for three-inch 
pitch, he uses that with the remark that it is 
"near enough," laying the blame on the 
odontograph or on the iron founder if the 
resulting gear roars. His next order is for a 



PITCH DIAMETERS. 
For 0-Ni: Inch Circular Pitch. 

FOR ANT OTHER PITCH MULTIPLY BY THAT PITCH. 



T. P. D. 



10 


3.18 


11 


3.50 


12 


3.82 


13 


4.14 


14 


4.46 


15 


4.78 


16 


5.09 


17 


5.41 


18 


5.73 


19 


6.05 


20 


6.37 


21 


6.69 


22 


7.00 


23 


7.32 


24 


7.64 


25 


7.96 


26 


8.28 


27 


8.60 


28 


8.91 


29 


9.23 


30 


9.55 


31 


9.87 


32 


10.19 



T. P. D. 



34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 



10.50 
10.82 
11.14 
11.46 
11.78 
12.10 
12.42 
12.73 
13.05 
13.37 
13.69 
14.00 
14.33 
14 64 
14.96 
15.28 
15.60 
15.92 
16.24 
16.55 
16.87 
17.19 
17.51 



T. P. D. 



17.83 
18.14 
18.46 
18.78 
19.10 
19.42 
19.74 
20.06 
20.37 
20.69 
21.01 
21.33 
21.65 
21.97 
22.28 
22.60 
22.92 
23.24 
23 56 
23.88 
24.19 
24.51 
24.83 



T. 


P.D. 


79 


25.15 


80 


25.47 


81 


25.79 


82 


26.10 


83 


26.42 


84 


26.74 


85 


27.06 


86 


27.38 


87 


27.70 


88 


28.01 


89 


28.33 


90 


28.65 


91 


28.97 


92 


29.29 


93 


29.60 


94 


29.92 


95 


30.24 


96 


30.56 


97 


30.88 


98 


31 20 


99 


31.52 


100 


31.83 



36. — THE DIAMETRAL PITCH. 

This is not a measurement, but a ratio gear. Thus, a gear of 48 teeth and 12 inches 



or proportion. It is the number of teeth in 
the gear divided by the pitch diameter of the 



pitch diameter is of 4 pitch. The advantages 
of the diametral pitch unit are so apparent 



16 



Pitches and Addendum. 



that it is fast displacing the circular pitch 
unit, and has almost entirely displaced it for 
cut gearing. It is so simple that a table of 
pitch diameters is entirely useless, although 
such useless tables have been published. 



The diametral pitch is sometimes denned as 
the number of teeth in a gear of one inch 
diameter. It is a common, but bad practice, 
to designate diametral pitches by numbers, 
as No. 4, No. 16, etc. 



37. — RELATION OF PITCH UNITS. 

The product of the circular pitch by the 
diametral pitch is the constant number 
3.1416, so that if one is given the other is 
easily calculated. 

The following tables of equivalent pitches 
will be convenient in this connection. 



38. — ACTUAL SIZES. 

Figs. 28 and 29 show the actual sizes of 
standard teeth of the usual diametral pitches, 
and give a better idea of the actual teeth than 
can be given by any possible description. 
They are printed from cut teeth, and may be 
depended upon as accurate. 



Diametral 


Circular 


Pitch. 


Pitch. 

1 


2 


1.571 inch 


2H 


1.396 " 


m 


1.257 " 


2M 


1.142 " 


3 


1.047 " 


m, 


.898 " 


4 


.785 " 


5 


.628 " 


6 


.524 " 


7 


.449 " 


8 


.393 " 


9 


.349 " 


10 


.314 " 


11 


.286 " 


12 


.262 " 


14 


.224 " 


16 


.196 kk 


18 


.175 " 


20 


.157 " 


22 


.143 " 


24 


.131 " 


26 


.121 " 


28 


.112 kt 


30 


.105 " 


32 


.098 " 


36 


.087 " 


40 


.079 kt 


48 


.065 " 



Circular 


Diametral 


Pitch. 


Pitch. 


2 


1 571 


1% 


1 676 


m 


1.795 


m 


1.933 


Wz 


2 094 


ill 


2 185 


2 285 


1t 5 f 


2 394 


m 


2.513 




2.646 


2.793 


1A 


2.957 


i 


3.142 


16 


3 351 


% 


3.590 


13 
IS 


3.867 


k 


4.189 


ii 
II 


4.570 


5.027 


9 

a 


5.585 


6.283 


I 


7 181 


8 378 


5 
15 


10 053 


li 


12.566 


3 

T5 


16.755 


% 


25.133 


1 


50.266 



39. — ADDENDUM AND DEDENDUM. 



The tooth is limited in length by the circle 
a I, Fig. 30, called the addendum line, and 
drawn outside the pitch line at a given 
distance, called the addendum. Its depth is 
also limited by aline r I, called the dedendum 
or root line, drawn at a given distance inside 
of the pitch line. 

The addendum and the dedendum are 
both arbitrary distances, but, for convenience 
in computation, they are fixed at simple 
fractions of the unit of pitch that is in use. 
When the circular pitch is used the ad- 
dendum is one-third of the circular pitch. 

When the diametral pitch unit is used 
the addendum is one divided by the pitch. 

It is customary to make the addendum and 
the dedendum the same, except in certain 
cases where some special requiremen is to 
be satisfied. 




Addenda 
Clearance 
Backlash 



Actual Sizes. 



17 




\& 



Actual Sizes. 



2 Pitch. 




Fig. 29. 



Items of Construction. 



10 



40.— THE CLEARANCE. 

To allow for the inevitable inaccuracies of The clearance, or distance of the clearance 
workmanship, especially on cast gearing, it line inside of the root line is arbitrary, but it 
is customary to carry the tooth space slightly , is convenient and customary to make it one- 
below the root line to the clearance line c I, j eighth of the addendum. 
Fig. 30. 



41. — THE BACK-LASH. 



When rough wooden cogs or cast teeth are 
used, the irregularities of the surface, and 
inaccuracies of the shape and spacing of 
the teeth, require that they should not pre- 
tend to fit closely, but that they should clear 
each other by an amount b, Fig. 30, called 
the back-lash. 

The amount of the back-lash is arbitrary, 



but it is a good plan to make it about equal 
to the clearance, one-eighth of the addendum. 
Skillfully made teeth will require less 
back-lash than roughly shaped teeth, and 
properly cut teeth should require no back- 
lash at all. Involute teeth require less back- 
lash than cycloidal teeth. 



42. — THE STANDARD TOOTH. 



The tooth must be composed o* odontoids, 
preferably of odontoids of which the proper 
ties are well known, and an advantage is 
gained if it is still further confined to a par- 
ticular value of that odontoid. If the teeth 
are to be drawn by an odontograph some 
standard must be fixed upon, since the 



method will cover but one proportion of tooth. 
For example, the standard involute tooth 
is that having its line of action inclined at an 
angle of obliquity of fifteen degrees. For 
the cycloidal system the standard agreed upon 
is the tooth having radial flanks on a gear of 
twelve teeth. 



43. — ODONTOGRAPHS. 



The construction of the tooth is generally 
not simply accomplished by graphical means, 
as it is generally required to find points in 
the curve and then find centers for circular 
arcs that will approximate to the curve thus 
laid out. 

It is sometimes attempted to construct the 
curve by some handy method or empirical 



rule, but such methods are generally worth- 
less. 

An odontograph is a method or an instru- 
ment for simplifying the construction of the 
curve, generally by finding centers for ap- 
proximating circular arcs without first find- 
ing points on the curve, and those in use will 
be described. 



44. — THE 
When the teeth are laid out by theory 
there will be a portion of the tooth space at 
the bottom that is never occupied by the 
mating tooth. Fig. 31 shows a ten-toothed 
pinion tooth and space with a rack tooth in 
three of its positions in it, showing the un- 
used portion by the heavy dotted line. 

If this unused space is filled in by a 
" fillet " y tb.e tooth, will be strengthened just 
where it needs it the most, at the root. 

The fillet is dependent on the mating tooth, 
and is therefore not a fixed feature of the 
tooth. If a gear is to work in an inter- 
changeable set, it may at some time work 
with a rack, and therefore its fillet should be 
fitted to the rack ; but if it is to work only 



FILLET. 

with some one gear it may be fitted to that. 
The light dotted line shows the fillet that 
would be adapted to a ten-toothed mate. 
The fillet to match an internal gear tooth 
would be even smaller than that made by the 
rack. 



Fig* 31* 




The fillet 



20 



Equidistant Series. 



When the tooth is formed by the molding 
process of (27), or by the equivalent planing 
process of (28), the fillet will be correctly 
formed by the shaping tool, but not so when 
the linear process of (29) is used. When the 
tooth is drawn by theory or by an odonto- 
graph the fillet must be drawn in, and can be 



most easily determined by making a mating 
tooth of paper, and trying it in several posi- 
tions in the tooth space, as in the figure. 

Except on gears of very few teeth the 
strength gained will not warrant the trouble 
of constructing the fillet. 



45. — THE EQUIDISTANT SERIES. 



When arranging an odontograph for 
drafting teeth, or a set of cutters for cutting 
them, we must make one sizing value do 
duty for an interval of several teeth, for it is 
impracticable to use different values for two 
or three hundred different numbers of teeth. 
The object of the equidistant series is to so 
place these intervals that the necessary errors 
are evenly distributed, each sizing value 
being made to do duty for several numbers 
each way from the number to which it is 
fitted, and being no more inaccurate than any 
other for the extreme numbers that it is 
forced to cover. 

This series is readily computed for any 
case that may arise, and with a degree of ac- 
curacy that is well within the requirements 
of practice; by the formula 
a n 



t = 



n — s -J- 



a s 



in which a is the first and z is the last tooth 
of the interchangeable series to be covered; 
n is the number of intervals in the series, and 
s is the number in the series of any interval 
of which the last tooth t is required. 

For example, it is required to compute the 
series here used for the cycloidal odonto- 
graph, having twelve tabular numbers to 
cover from twelve teeth to a rack. 

Putting a = 12, z = infinity, and n = 12, 
the formula becomes 

12 X 12 12 X 12 144 



t = 



00 



12 — s-J-0 12— 8 



and then, by putting s successively equal to 
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12, we get 
the series of last teeth, 13 r V, 14|, 16, 18, 20£, 
24, 28|, 36, 48, 72, 144, and infinity. These 
give the required equidistant series of inter- 
vals. 



12 
13 to 14, 
15 to 16, 
17 to 18, 
19 to 21, 
22 to 24, 



25 to 29 
30 to 36, 
37 to . 48, 
49 to 72, 
73 to 144, 
145 to a rack ; 
and the method is as easily applied to any 
other practical example. 

This formula and method is independent of 
the form and of the length of the tooth, and 
therefore is applicable to all systems under 
all circumstances. This is proper and con- 
venient, for these elements can be eliminated 
without vitiating the results or destroying the 
"equidistant" characteristic of the series. 
The formula is an approximation based upon 
an assumption, but nothing more convenient 
or more accurate has so far been devised bv 
laboriously considering all the petty elements 
involved. 

The sizing value, or number for which the 
tabular number is computed, or the cutter is 
accurately shaped, can best be placed, not at 
the center of the interval, but by considering 
the interval as a small series of two intervals, 
and adopting the intermediate value. 'J he 
sizing value for the interval from c to d is 

given by the formula 

2 cd 

v = 



c -f- d ' 

Thus, the sizing value for the interval 
from 37 to 48 teeth should be 41.8, and that 
for the interval from 145 to a rack should be 
290. 

It is sometimes the practice to size the cut- 
ter for the lowest number in its interval, on 
the ground that a tooth that is considerably 
too much curved is better than one that is 
even a little too fiat. This makes the last 
tooth of the interval much more inaccurate 
than if the medium number was used- 



Friction of Approach. 



21 



46. — THE HUNTING COG. 



It is customary to make a pair of cast gears 
with incommensurable numbers of teeth so 
that each tooth of each gear will work with 
all the teeth of the other gear. If a pair of 
equal gears have twenty teeth each, each 
tooth will work with the same mating tooth 
all the time; but if one gear has twenty and 
the other twenty-one teeth, or any two num- 
bers not having a common divisor, each tooth 
will work with all the mating teeth one after 
the other. 

The object is to secure an even wearing 
action; each tooth will have to work with 
many other teeth, and the supposition is that 



all the teeth will eventually and mysteriously 
be worn to some indefinite but true shape. 

It would seem to be the better practice to 
have each tooth work with as few teeth as 
possible, for if it is out of shape it will dam- 
age all teeth that it works with, and the 
damage should be confined within as narrow 
limits as possible. If a bad tooth works with 
a good one it will ruin it, and if it works 
with a dozen it will ruin all of them. It is 
the better plan to have all the teeth as near 
perfect as possible, and to correct all evident 
imperfections as soon as discovered. 



47. — THE MORTISE WHEEL. 

Another venerable relic of the last century 
is the "mortise" gear, Fig. 32, having 
wooden teeth set in a cored rim, in which 
they are driven and keyed. 

Where a gear is subjected to sudden strains 
and great shocks, the mortise wheel is better, 
and works with less noise than a poor cast 
gear, and will carry as much as or more 
power at a high speed with a greater dura- 
bility. But in no case is it the equal of a 
properly cut gear, while its cost is about as 
great. 

In times when large gears could not be cut, 
and when the cast tooth was not even ap- 
proximately of the proper shape, the mortise 
wheel had its place, but now that the large 
cut gear can be obtained the mortise gear 
should be dropped and forgotten. 




JUortise wheel 

Fig. 32. 



48. — THE FRICTION OF APPROACH. 



When the point of action between two 
teeth is approaching the pitch point, that is, 
when the action is approaching, the friction 
between the two tooth surfaces is greater than 
when the action is receding. This extra fric- 
tion is always present, but is most trouble- 
some when the surfaces are very rough, as on 
cast teeth, giving little trouble when the teeth 
are properly shaped and well cut. When the 
roller pin gear (93) is used, the friction 
between the teeth is rolling friction, and is 
no greater on the approach than on the recess. 




tion of 
roach 



Fig. 33. 



22 



Efficiency . 



The difference in the friction is probably- 
due to the difference in the direction of the 
pressure between the small inequalities to 
which all friction is due. When the gear D, 
Fig. 33, is the driver, the action between the 
teeth is receding, and the inequalities lift over 
each other easily, while if F is the driver, 
the action is approaching, and the inequalities 
tend to jam together. 

In the exaggerated case illustrated, it is plain 
that the teeth are so locked together that ap- 
pro idling action is impossible, while it is 
equally plain that motion in the other direc- 
tion is easy. The same action takes place in 
a lesser degree with the small inequalities of 
ordinary rough surfaces. 



The action of the common friction pawl, 
which works freely in one direction and jams 
hard in the other, is upon the same principle. 
A weight may be easily dragged over a rough 
surface that it could not be pushed over by a 
force that is not parallel to the surface. 

The extra friction of approaching action 
can be avoided by giving the driver the long- 
est face. When the driver has faces only, 
and the follower has only flanks, the action is 
particularly smooth. 

Teeth that are subject to excessive maxi- 
mum obliquity, such as cycloidal teeth, should 
not be selected for rough cast gearing, for it 
is the maximum rather than the average obli- 
quity that has the greatest influence. 



49. — EFFICIENCY OF GEAR TEETH. 



Much has been written, but very little has 
been done to determine the efficiency of the 
teeth of gearing in the transmission of power, 
and therefore but little of a definite nature 
can be said. The question is mostly a prac- 
tical one, and should be settled by experi- 
ment rather than by analysis. 

The only known experiments upon the fric- 
tion of spur gear teeth are the Sellers experi- 
ments, more fully detailed in (112), and but. 
one of these relates to the spur gear. From 
that one it is known that a gear of twelve 
teeth, two pitch, working in a gear of thirty- 
nine teeth, has an efficiency varying from 
ninety per centum at a slow speed to ninety- 
nine per centum at a high speed. That is, 
an average of five per centum of the power 
received is wasted by friction at the teeth and 
shaft bearings. This result is probably a 
close approximation to that for any ordinary 
practical case. 

Although theory can do nothing to de- 
cide such a question as this, it can do much 
to indicate probable results. 

If a pair of involute teeth, for example, 
move over a certain distance, w, either way 
from the pitch point, the distance being mea- 
sured on the pitch line, they will do work that 
is theoretically determined by the formula : 

k ± h 



f P 
work done = ■ -g- 

a 



k h 



w 



in which / is the coefficient of friction, P is 



the pressure, and k and h are the pitch radii 
of the gears. The positive sign is to be used 
for gears in external, and ihe negative sign 
for those in internal contact. 

The loss by friction, as shown by the for- 
mula, decreases directly as the diameters in- 
crease, the proportion of the diameters being 
constant. 

The loss increases rapidly with the distance 
of the point of action from the pitch point. 
When the contact is at the pitch point the 
teeth do not slide on each other, and there is 
no loss, but away from that point the loss is 
as the square of the distance in this case, and 
in a still greater proportion in the case of the 
cycloidal tooth. Therefore a short arc of 
action tends to improve the efficiency. 

It has been satisfactorily determined that 
the loss is greater during the approaching 
than during the receding action. This is not 
shown by the formula, but it may be laid to 
a variation in the coefficient /. 

The formula shows that the loss is inde- 
pendent of the width or face of the gear, 
and therefore strength can be increased by 
widening the face, without increasing the 
friction. 

If the work of internal gearing is com- 
pared with that of external gearing of the 
same sizes, the losses are in the proportion, 
k— h 
k-\- h'' 



Strength. 



23 



so that the internal gear is much the more 
economical, particularly when the gear and 
pinion are nearly of the same size. If the 
gear is twice the size of the pinion the loss 
is but one-third of the loss when both gears 
are external. 

Small improvement can oe effected, by put- 
ting a small pinion inside rather than outside 
of a large gear. A six-inch pinion working 
with a six-foot gear has but 1.18 times the 
loss by the same gears, when the gear is in- 
ternal. 

Theoretical efficiency is discussed at great 
length in the Journal of the Franklin Insti- 
tute, for May, 1887. Also by Reuleaux, and 
again by Lanza, in the Transactions of the 



American Society of Mechanical Engineers 
for 1887, and the discussion has been carried 
far enough. 

A series of experiments with gear teeth of 
various sizes and forms, of various metals, 
would add greatly to our knowledge of this 
important matter. 

A true determination of the efficiency of 
the rough cast gear, as compared with that of 
the cut gear, would tend to discourage the 
use of the former for the transmission of 
power, for experiment would undoubtedly 
show that the power wasted by the cast gear 
would soon pay the difference in cost of the 
better article. 



5C. — STRENGTH OF A TOOTH. 



The strength of a tooth is the still load it will 
carry, suspended from its point, and is to be 
carefully distinguished from the horse-power, 
or the load the gear will carry in motion. 

The strength of a substance is not a fixed 
element, but will vary with different samples, 
and with the same sample under different 
circumstances ; allowance must be made for 
the amount of service the sample has seen, 
concealed defects must be provided against, 
and therefore nothing but an actual test will 
surely determine its character. 

Although no possible rule can be depended 
upon, the ultimate or breaking strength of a 
standard cast-iron tooth, having an addendum 
about equal to a third of the circular pitch, 
will average about three thousand five hun- 
dred pounds multiplied by the face of the 
gear and again by the circular pitch, both 
in inches. 

But a tooth should never be forced up to 
its ultimate strength, and the best practice is 
to give it only about one-tenth of tbe load it 
might possibly bear, so that the following 
rule should be used : Multiply three hundred 
and fifty pounds by the face of the gear, and 
again by the circular pitch, both in inches, 
and the product will be the safe working 
load of one tooth. 

Example : A cast-iron gear of one inch 
pitch, and two inches face, wiil safely lift 
350 X 2 x 1 = 700 pounds, although it 
would probably lift 7,000 pounds. 



When there are two teeth always in work- 
ing contact, it is safe to allow double the 
load, but care must be taken that both teeth 
are always in full contact. 

A hard wood mortised cog has about one- 
third of the strength of a cast-iron tooth ; 
steel has double the strength ; wrought-iron 
is not quite as strong. 

A small pinion generally has teeth that are 
weak at the roots, and then it will increase 
the strength to shroud the gear up to its 
pitch line, but shrouding will not strengthen 
a tooth that spreads towards its base, like an 
involute tooth, and when the face of the 
gear is wide compared with the length of the 
tooth the shroud is of rittle assistance. 

It does not increase the strength of a tooth 
to double its pitch, for when the pitch is 
increased the length is also increased, and the 
strength is still in direct proportion to the 
circular pitch, wnile the increase has reduced 
the number of teeth m contact at a time. 

Cut gears and cast gears are about equal 
as to actual strength, with the advantages in 
favor of the cut gear, that hidden d sfects are 
likely to be discovered, and that it is not as 
liable to undue strains on account of defective 
shape. 

The rules for strength must not be used for 
gears running at any considerable speed, for 
they are intended only for slow service, as in 
cranes, heavy elevators, power punches, etc. 



24 



Horse- Power. 



12 X 2 X 2 X 4 X a/ 24 X 100 



51. — HORSE-POWER OP CAST GEARS. 

The horse-power of a gear is the amount 
of power it may be depended upon to carry- 
in continual service. 

It is very well settled that continual strains 
and impact will change the nature of the 
metal, rendering it more brittle, so that a 
tooth that is perfectly reliable when new 
may be worthless when it has seen some years 
of service. This cause of deterioration is 
particularly potent in the case of rough cast 
teeth, for they can only approximate to the 
true shape required to transmit a uniform 
speed, and the continual impact from shocks 
and rapid variations in the power carried 
must and does destroy the strength of the 
metal. 

There are about as many rules for com- 
puting the power of a gear as there are 
manufacturers of gears, each foundryman 
having a rule, the only good one, which he 
has found in some book, and with which he 
will figure the power down to so many 
horses and hundredths of a horse as con- 
fidently as he will count the teeth or weigh 
the casting. 

Even among the standard writers on en- 
gineering subjects the agreement is no bet- 
ter, as shown by Cooper's collection of 
twenty-four rules from many different wri- 
ters, applied to the single case of a five-foot 
gear. See the "Journal of the Franklin 
Institute" for July, 1879. For the single 
case over twenty different results were ob- 
tained, ranging from forty-six to three- 
hundred horse-power, and proving conclu- 
sively that the exact object sought is not to be 
obtained by calculation. 

This variety is very convenient, for it is 
always possible to fit a desired power to 
a given gear, and if a badly designed gear 
should break, it is a simple matter to find a 
rule to prove that it was just right, and must 
have met with some accident. 



Although no rule can be called reliable, 
the one that appears to be the best is that 
given by Box, in his Treatise on Mill Gear- 
ing. Box's rule, which is based on many 
actual cases, and which gives among the 
lowest, and therefore the safest results, is by 
the formula: 

Horse-power of a cast gear = ^-~ : 

v 6 1,000 

in which c is the circular pitch, /is the face, 

d is the diameter, all in inches, and n is the 

number of revolutions per minute. 

Example : A gear of two feet diameter, 

four inches face, two inches pitch, running 

at one hundred revolutions per minute, will 

transmit 



= 9.4 h. p. 



1,000 

For bevel gears, take the diameter and 
pitch at the middle of the face. 

It is perfectly allowable, although it is not 
good practice, to depend upon the gear for 
from three to six times-the calculated power, 
if it is new, well made, and runs without 
being subjected* to sudden shocks and varia- 
tions of load. 

The influence of impact and continued 
service will be appreciated when it is con- 
sidered that the gear in the example, which 
will carry 9.4 horse-power, will carry seventy 
horse-power if impact is ignored, and the 
ultimate strength of the metal is the only 
dependence. 

A mortise gear, with wooden cogs, will 
carry as much as, or more than a rough cast- 
iron gear will carry, although its strength is 
much inferior. The elasticity of the wood 
allows it to spring and stand a shock 
that would break a more brittle tooth of 
much greater strength. And, for the same 
reason, a gear will last longer in a yielding 
wooden frame than it will in a rigid iron 
frame. 



52. — HORSE-POWER OP CUT GEARS. 

We know a little, and have to guess the data upon which a reliable rule can be 



rest, as to the power of a cast gear, but with 
respect to that of a cut gear we are not as 
well posted, for there are no experimental 



founded. 

Admitting, as we must, that impact is 
the chief cause of the deterioration of the 



The Involute Toot/2. 



25 



cast gear, we are at liberty to assume that a 
properly cut and smoothly running cut gear 
is much more reliable. 

No definite rule is possible, but we can 
safely assume that a cut gear will carry at 
least three times as much power as can be 
trusted to a cast gear of the same size. 

The great reliance of those who claim that 
a cast gear is superior to a cut gear is upon 
the hard scale with which the cast tooth is 
covered. This scale is not over one-hun- 
dredth of an inch thick, is rapidly worn 



away, and is of no account whatever. From 
that point of view it is difficult to explain 
why a wooden tooth will outwear an iron one, 
although it is softer than the softest cut iron. 
Assuming that a cut gear is about three 
times as reliable as a cast gear, we can com- 
pute its power by the formula : 

Horse-power of a cut gear = — — ^ 

ov 

in which c is the circular pitch, / is the face, 

and d is the pitch diameter, all in inches, and 

n is the number of revolutions per minute. 



3. THE INVOLUTE SYSTEM. 



S3. — THE INVOLUTE TOOTH. 

The simplest and best tooth curve, theo- 
retically, as well as the one in greatest prac- 
tical use for cut gearing, is the involute. 

The involute tooth system is based on the 
straight rack odontoid, (31) and Fig. 21, and 
it is illustrated by Fig. 34. If the four odon- 
toids of the rack outline are equally inclined 
to the pitch line, the resulting tooth system 
will be completely interchangeable; but if, 
as in Fig. 35, the face and flank are inclined 
at different angles of obliquity, T 8 K and 
T S K' , the system is not interchangeable, 
although otherwise perfect. 

The rack odontoid cannot have a corner or 
change of direction anywhere except at the 
pitch line; without causing a break in the 
line of action. 

As the normals p q are parallel, the line of 
action is a straight line WO W at right 
angles to the rack odontoid. The inter- 
changeable line of action is continued in a 
straight line on both sides of the pitch line, 
but the non- interchangeable line changes di- 
rection at that line. 

In accordance with the universal custom 
we will consider that the involute tooth is 
always interchangeable, having a single angle 
ot obliquity. 




Fig. 



The involute tooth 
interchangeable 

b 

3d. 




^on-interchangeable 
involute 

Fig. 35* 



20 



hivolute Interference. 



54. — THE CUSP. 



As a circle i c, Fig. 34, can always be drawn 
tangent to the line of action at an interfer- 
ence point i, from the center b of any pitch line 
B, there will always be a cusp in the curve at 
the point c (16), and at that point the working 
part of the curve must stop. The working 
part of the rack tooth must end at the limit 
line i L through the interference point i. 

The working curves of any two teeth that 
work with each other must each end at the 
line drawn through the interference point of 
the other, Fig. 43, being limited by limit 
lines 1 1 and L L. 



The second branch c m! of the curve is 
equal to the first branch c m, but is re- 
versed in direction. The second cusp is at 
infinity, and therefore has no practical ex- 
istence. 

The tangent circle i c, through the inter- 
ference point and the cusp, is called the 
"base line." 

It is customary to continue the flank of the 
tooth inside the base line by a straight radial 
line, as far as may be necessary to allow the 
mating gear to pass. 



55 . — IN TERFERENCE . 

When the point of the tooth is continued 
beyond the limit line it will interfere with and 
cut away a portion of the working curve of 
the mating tooth. Fig. 36 shows a rack tooth 
working with the tooth of a small pinion, and 
cutting out its working curve. 

This cut is not confined to the flank, but 
extends across the pitch line into the face, as 
shown by the line qmn. The rack tooth of 
the figure will not work with the pinion tooth 
unless it is cut off at the limit line 1 1 through 
the interference point i. 

The mathematical action still continues, 
and the figure shows the rack tooth in action 
at h with the second branch of the curve. 




Effect of Interference 

Fig. 3tf. 



56. — ADJUSTABILITY. 



An interesting and in many cases a valua- 
ble feature of the involute curve, and one 
that is confined to it, is the fact that its posi- 
tion as a whole with regard to the mating 
curve is adjustable. 

Two involutes, each with its base line, will 
work together in perfect tooth contact when 
they are moved with respect to each other, 
as long as they touch at all. The lines of 
action and the pitch lines will shift as the 
curves are moved, and will accommodate 
themselves to the varying position of the 
base lines. 

But this valuable feature of the involute 
curve is not always available, and involute 
gears are not, as commonly supposed, neces- 



sarily adjustable, for the conditions are often 
such that the teeth will fail to act when the 
centers are moved, except within very narrow 
limits. Care must be taken that the arc of 
action is not so reduced by separating the 
centers of the gears that it is less than the cir- 
cular pitch, for the former arc is variable and 
the latter is fixed. Care must also be taken 
that the working curve is not pushed over the 
limit line when the centers are drawn to- 
gether. 

In any limiting case, such as in Fig. 43, the 
centers are not adjustable. The gears of the 
standard set are either not adjustable at all 
or are so within very narrow limits, on ac- 
count of the correction for interference. 



Involute Construction. 



27 



57.— CONSTRUCTING THE INVOLUTE BY POINTS. 



The simple involute curve can be con- 
structed by points by the general method of 
(24), but it is much better to take advantage 
of the property that it is an involute of its 
base circle, and construct it by the rectifica- 
tion of that circle. 

As in Fig. 37 any convenient small dis- 
tance A G is taken on the dividers, and the 
points on the curve located by stepping 
along the circle and its tangent from any 
given point to any desired point. 

This method is so accurate, if care is taken 
to step accurately on the line, that the curve 
seldom needs correction; but, when great ac- 
curacy is required, correction can be applied 
at the rate of one- thousandth of an inch to 
the step, if the length of the step is regulated 
by the diameter of the circle according to the 
following table: 

Diameter of Circle : 
123456789 10 11 12 



Length of Step : 
.17 .26 .37 .46 



53 .60 .67 .73 .76 .79 .82 .84 



For example: If the circle of Fig. 37 is 




Construction by 
points 

Fig. 37. 



four inches in diameter, and the dividers are 
set to .46 inch, the true curve, A V d', will be 
outside of the constructed curve A b d by .002 
inch at b and .005 inch at d. 

From the table we can form the handy and 
sufficiently accurate rule that the length of 
the step should be about one-tenth of the di- 
ameter of the circle, for a correction of about 
one-thousandth of an inch per step. 

Having thus found several points of the in- 
volute, we can draw it in by hand, or by con- 
structing a template, or by finding centers 
from which approximately accurate circular 
arcs can be drawn. 



58. — THE STANDARD INVOLUTE TOOTH. 



The tooth that is selected for general use, 
and the one that is the best for all except a 
few special cases and limiting cases, is the in- 
terchangeable tooih having an angle of ob- 
liquity of fifteen degrees, an addendum of 
one-third the circular pitch, or one divided by 
the diametral pitch, and a clearance of one- 
eighth of the addendum. 

The standard to which involute cutters are 
made is slightly different, having an angle of 
14° 28' 40", the sine of which is one-quarter, 
and a clearance of one-twentieth of the circu- 
lar pitch. 

If the obliquity is 15° the smallest possible 
pair of equal gears have 11.72 teeth, and 



therefore 12 is the smallest gear of the inter- 
changeable set. 

The base distance, the distance of the base 
line inside of the pitch line, is about one-fifty- 
ninth of the pitch diameter, and one-sixtieth 
is a convenient fraction for practical use. 

The limit points of the whole set must be 
determined by that of the twelve-toothed 
gear, for any gear of the set may be required 
to work with that one, and the working curve 
of each tooth must end at the point thus de- 
termined. As the limit point is always in- 
side of the addendum line there must always 
be a false extension on the tooth, the point 
being rounded over outside of the limit point. 



59. — THE INVOLUTE ODONTOGRAPH. 



As the base line must always be drawn, it 
is advisable, to save work, to locate the cen- 
ters of the approximate circular arcs upon 
that line. It is also necessary that the points 
of the teeth shall be rounded over, to avoid 



interference. These requirements made it 
impracticable to compute the positions of the 
centers, and an empirical rule had to be adopt- 
ed instead. 
Teeth were carefully drawn by the stepping 



28 



Ten and Eleven Involute Teeth. 



method of (57) on a very large scale, one- 
quarter pitch, giving a tooth eight inches in 
length. These teeth were corrected for inter- 
ference by giving them epicycloidal points 
that would clear the radial flanks of the 
twelve-toothed pinion. 

Then the proper centers on the base line 
were determined by repeated trials, and tooth 
curves obtained that would agree with the 
true involute up to the limit point, and still 



clear the corrected point. The odontograph 
table is a record of these radii, which are be- 
lieved to be as nearly correct as the given 
conditions will permit. 

It was found that separate curves were 
required for face and flank up to thirty-six 
teeth, but that one curve would answer for 
teeth beyond. 

It was found necessary to devise a separate 
method for drafting the rack tooth. 



Theoretically the twelve-toothed pinion is 
the smallest standard gear that will have an 
arc of action as great as the circular pitch, 
but ten and eleven teeth may be used with 
an error that is not practically noticeable. 
Fig. 38 shows a pair of ten-toothed gears in 



60. — TEN AND ELEVEN TEETH. 

action. They can be in correct action only 
when the point of contact is between the two 
interference points % and I, but they will be 
in practical contact for a greater and suffi- 
cient distance 




Fig. 3S. 



Odontographic pair 



61. — A BAD KULE. 



There is a simple and worthless rule for 
involute teeth that deserves notice only be- 
cause it is considerably in use. 

It constructs the whole tooth curve, face and 



flank, for all numbers of leeth, as a single convenience is purchased at the expense of 



arc from a center on the base line, and with a 
radius equal to one-quarter of the pitch radius, 
Fig. 39. 
This is wonderfully convenient, but the 



The Involute Odontograph. 



29 



ordinary accuracy, for the rule is not even 
approximately correct. It is handy, and 
nothing else. 

Figs. 38 and 40 show the kind of teeth 
that are constructed by this rule on gears of 
ten and twelve teeth, where its error is the 
greatest, and it is reasonable that the invo- 
lute tooth should not be in great favor with 
those who have been taught to draw it thus. 

The error gradually decreases, until, for 
more than thirty teeth, it is tolerably correct, 
but it gives the rack with the straight, uncor- 
rected working face that would interfere, as 
shown at q, Fig. 40., 

As it is tolerable only for thirty or more 



teeth, and not good then, it may well be 
dropped altogether. 




A. bud rule 

Fig. 39. 



62. — USING THE INVOLUTE ODONTOGRAPH. 

INVOLUTE ODONTOGRAPH. 
Standard Interchangeable Tooth, Centers on Base Line. 
{For Table of Pitch Diameters see 35.) 





Divide by the 


Multiply by the 




Diametral Pitch. 


Circular Pitch. 


Teeth. 












Face 


Flank 


Face 


Flank 




Radius. 


Radius. 


Radius. 


Radius. 


10 


2.28 


.69 


.73 


.22 


11 


2.40 


.83 


.76 


.27 


12 


2.51 


.96 


.80 


.31 


13 


2.62 


1.09 


.83 


.34 


14 


2.72 


1 22 


.87 


.39 


15 


2 82 


1.34 


.90 


.43 


16 


2.92 


1.46 


.93 


.47 


17 


3.02 


1.58 


.96 


.50 


18 


3.12 


1.69 


.99 


.54 


19 


3.22 


1.79 


1.03 


.57 


20 


3.32 


1.89 


1.06 


.60 


21 


3.41 


1.98 


1 09 


.63 


22 


3.49 


2.06 


1.11 


.66 


23 


3.57 


2 15 


1.13 


.69 


24 


3.64 


2.24 


1.16 


.71 


25 


3.71 


2.33 


1 18 


.74 


26 


3.78 


2.42 


1.20 


.77 


27 


3.85 


2.50 


1.23 


.80 


28 


3 92 


2.59 


1.25 


.82 


29 


3 99 


2.6T 


1.27 


.85 


30 


4.06 


2.76 


1.29 


.88 


31 


4.13 


2.85 


1.31 


.91 


32 


4.20 


2.93 


1.34 


.93 


33 


4.27 


3 01 


1 36 


.96 


34 


4.33 


3 09 


1.38 


.99 


35 


4.39 


3 16 


1.39 


1.01 


36 


4.45 


3 23 


1.41 


1.03 


37^0 


4.20 
4.63 


1.34 


41-45 


1.48 


46—51 


5 06 


1.61 


52-60 


5 74 


1.83 


61-70 


6.52 


2.07 


71 90 


7.72 - 


2.46 


91-12") 


9.78 


3 11 


121-180 


13 38 


4.26 


181-360 


21 


62 


6. 


88 



Draw the rack tooth by the special method. 



30 



The Involute Odontograph. 



To draft the tooth lay off the pitch, ad- 
dendum, root, and clearance lines, and space 
the pitch line for the teeth, as in Fig. 40. 

Draw the base line one-sixtieth of the pitch 
diameter inside the pitch line. 

Take the tabular face radius on the divid- 
ers, after multiplying or dividing it as re- 
quired by the table, and draw in all the faces 



from the pitch line to the addendum line 
from centers on the base line. 

Set the dividers to the tabular flank radius, 
and draw in all the flanks from the pitch line 
to the base line. 

Draw straight radial flanks from the base 
line to the root line, and round them into the 
clearance line. 




Odontoffraphic example 



63. — SPECIAL RULE FOR THE RACK. 



Draw the sides of the rack tooth, Fig. 40, 
as straight lines inclined to the line of centers 
c c at an angle of fifteen degrees, best 
found by quartering the angle of sixty de- 
grees. 

Draw the outer half a b of the face, one- 



quarter of the whole length of the tootn, 

from a center on the pitch line, and with a 

radius of 

2.10 inches divided by the diametral pitch. 

.67 inches multiplied by the circular pitch. 



64. — DRAFTING INTERNAL GEARS. 



When the internal gear is to be drawn, the 
odontograph should be used as if the gear 
was an ordinary external gear. See Fig. 41. 

But care must be taken that the tooth of 
the gear is cut off at the limit line drawn 
through the interference point i of the pin- 
ion. The point of the tooth may be left off 



altogether or rounded over to get the appear- 
ance of a long tooth. 

The pinion tooth need not be carried in to 
the usual root line, but, as in the figure, may 
just clear the truncated tooth of the gear. 

The curves of the internal tooth and of its 
pinion may best be drawn in by points (57), 



The Involute Odontografth, 



31 



for the odontography corrected tooth is not 
as well adapted to the place as the true tooth, 
and no correction for interference is needed 
on the points of the pinion teeth or on the 
flanks of those of the gear. 



Care must be taken that the internal teeth 
do not interfere by the point a striking the 
point b, as they will if the pitch diameters 
are too nearly of the same size. 




Internal involutes 



W'41. 



65. — INVOLUTE GEARS FOR GIVEN OBLIQUITY AND ADDENDA. 



When the obliquity and addenda, as well 
as the pitch diameter and number of teeth in 
a gear arc given, as is generally the case, we 
can proceed to draft the complete gear as 
follows: 

Draw the pitch line p I, Fig. 42, the ad- 
dendum line a I, the root line r I, and the 
clearance line c I, as given. Draw the line of 
action I a at the given obliquity W Z = K. 
Draw the base line b I tangent to the line of 
action. Find the interference point * by bi- 
secting the chord v. 

Draw the involutes i am and i" a" m" , 
and a a" will be the maximum arc of ac- 
tion. 

If the given arc of action a a' is not great- 
er than the maximum arc, the pitch line is 
to be spaced and the tooth curves drawn in 
from the base line to the addendum line. 



These tooth curves, when small, are best 
drawn as circular arcs from centers on or 
near the base line, one center x for the flank 
from the base line to the pitch line, and 
another center y for the face from the pitch 
line to the addendum line. One involute 
i a m should be carefully constructed by 
points, and then the required ceuters can be 
found by trial. One center and arc will 
often answer for the whole curve, and it is 
only when great accuracy is required that 
more than two centers will be necessary. 

Continue the flanks of the teeth toward 
center by straight radial lines, and round 
these lines into the clearance line. 

If the interference point for the gear that 
the gear being drawn is to work with is at I, 
within the addendum line, the limit line 1 1 
must be drawn through it, and the points of 



32 



Involute Special Cases. 




Given obliquity and addendum 



the teeth outside of this limit must be slightly 
rounded over, to avoid interference (55). 
If a fillet / is desirable, to strengthen the 



tooth, it can be drawn in by the method 
of (44). 



6G. — INVOLUTE GEAKS FOE. GIVEN NUM- 
BERS OP TEETH. 

When the numbers of teeth and the pitch 
lines are the only given details, the shape and 
action of the tooth depends upon the obli- 
quity, and the action will fail if the angle is 
too small. The principal object is to deter 
miDe the least possible angle that is permitted 
by the given pitch diameters and numbers of 
teeth. 

Draw the pitch lines P L and p I, Fig. 43, 
lay off the given pitch arc, as a straight line 
c d or G D, at right angles to the line of 
centers, and draw the line C d or c B. Then 
the required line of action will be I a pac- 
ing through at right angles to c D or C d. 
The complete teeth can then be drawn in as 
previously directed. 

In this case, the obliquity W Z being the 
least possible, the limit lines and the adden- 
dum lines must coincide, but the addenda 
may be reduced by increasing the angle. 




t e ai 



&• al 



Given numbers of teeth 



Limiting- Involute Teeth. 



33 



67. — INVOLUTE GEAKS FOR GIVEN OBLIQUITY. 



When the pitch diameters and the obliquity- 
are the only given details, the lines C I and 
c i, Fig. 43, drawn from the centers at right 
angles to the line of action, will determine 
the limit lines. The maximum arc of action 
a a' may be found either by drawing the 
involutes % a and la', or by continuing the 
line C I to the line c d, and measuring the 



required distance c d. Any arc of action less 
than a a' may be used. 

The drawings should always be made to 
a scale of one tooth to the inch radius, so 
that the pitch arc will be %n. If the scale 
is one tooth to the inch of diameter, the 
pitch arc will be ?r. 



68. — INVOLUTE GEABS WITH LESS THAN FIVE EQUAL TEETH. 



The method of Fig. 43 and (66) will be 
found to apply to' any given numbers of 
teeth not less than five, and to fail, if either 
gear has but three or but four teeth. Any 
external gear of five or more teeth will work 
with any external gear of five or more teeth, 
and with an internal gear of any number of 
teeth unless stopped by internal interfer- 
ence (64). 

For example, if a pair having four and five 
teeth, Fig. 44, is tried, the four-toothed 
pinion will fail, because its tooth will come 
to a point upon the line of action before it 
has passed over the required pitch arc. The 
difficulty cannot be remedied by increasing 
the obliquity, for an angle that would allow 
the four-toothed pinion to act would also 
cause the five-toothed pinion to fail. 

The practical limit is five teeth, but the 
mathematical limit is the pair having the 
fractional number 4.62 teeth, Fig. 45. 

The four-toothed pinion will not work with 
any external gear, not even with a rack, but 
it will work with an internal gear that has 
about ten thousand teeth, and is practically a 
rack. It will work with any internal gear 
having less than ten thousand teeth, and Fig. 
46 shows it working with an internal gear of 
six teeth. Internal interference will prevent 
its working with an internal gear of five 
teeth. 

The three-toothed pinion has no practical 
action. It has a mathematical action with in- 
ternal gears of 3.56 or less teeth, as shown 
by Fig. 47, but as its limit is less than four, it 
cannot work with any whole number. The 
figure shows the interference at a. 

The extreme mathematical limit may be 
said to be the gear of 2.70 teeth, which has a 



theoretical action with an internal gear of the 
same size, coinciding with it. 




4.G2 



X 4.62 limit for equal tvdh 

Fig. 45. 



Limiting Involute Teeth, 




69. — INVOLUTE GEARS WITH LESS THAN 
FIVE UNEQUAL TEETH. 

If we drop the condition that the pitch 
line must be equally divided into tooth and 
space arcs, we can make gears of three and of 
four teeth work with external gears by the 
method of (65). The failing case of Fig. 44 
may be corrected by widening the failing 
tooth until it acts, and narrowing the other 
tooth to correspond, as shown in broken lines. 

In this way a four-toothed pinion will 
work with any number of teeth not less than 
5.57, at which limit both gears have pointed 
teeth, as in Fig. 48. 

The three-toothed pinion will work with 
any gear having 10.17 or more teeth. Fig. 49 
shows the 3x10.17 limiting pair, and Fig. 50 
shows the three-toothed pinion workiDg 
with an internal gear of five teeth. It will 
not work with an internal gear of four teeth, 
on account of internal interference, and there- 
fore the combination shown by Fig. 50 may 
be said to be the least possible symmetrical in- 
volute pair. 

A. gear of 2.70 teeth will work with a rack, 
but there seems to be no way to make a 
pinion of two teeth work under any circum- 
stances. 




4 teetU 

Fig. 4:8, 



Limiting' Involute Teeth. 



35 



Fig. 49* 

10.17 teetU 




3 X 10.17 
Unequal teeth 



Fig. 50 




70. — THE MATHEMATICAL LIMITS. 



The above results for low numbered pinions 
can be obtained by graphical means, but that 
method is not accurate enough to determine 
the limits with great precision, and in any 
case is tedious and laborious. 

The mathematical process is not particu- 
larly difficult, and consists in repeated trials 
with given formulae. 

To determine the obliquity at which a 
limiting pinion will be pointed on the line of 
action, for tooth equal to space, we use the 
formulae : 

2tt M 



tan. h = 



n{M+n) + ±n* 
4 M h n _ 1 



M +n 90 
in which n is the given number of teeth in 
the pointed gear, Fig. 51, M is the number 
in the gear having the radius M, and h is 
the angle c I. Knowing n, we assume a 
value for M, and from that find a value 
for h by means of the first formula. This 
value of h, tried in the second formula, will 
give an error. A second assumption for M 
will give a second error, and if the two 
errors are not too great a comparison will 
nearly locate the true value of M. 
Knowing n and M, we find the obliquity 

from 

2tt 



tan. K = 



M+n 






pointed 


pinion. 






Fig. 


51, 


> 


In this way 


the 


follov 


ring values were de- 


termined : 










n 




M 




K 


2.695 




1.26 




57° 49' 


3. 




1.51 




54° 20' 


4. 




2.86 




42° 29' 


4.62 




4.62 




34° 11' 


5. 




6.75 




28° 8' 


5.58 




00 








Having determined the obliquity for the 
pointed pinion, we can determine the least 
number of teeth it will work with by means 
of the following formulae : 

. - „ 180 n ^ „ 90 . „ 
Angle B — = tan. K ^ + K 

tan. B = -^r tan. K-\- tan. K 
in which N is the required least number. 



36 



Limiting Involute Teeth. 



In this way it was found that a gear of four 
teeth will not work with a racK, but will work 
with an internal gear having a number of 
teeth not easily calculated with existing loga- 
rithmic tables, but which is approximately 
ten thousand. Also that a pinion of three 
teeth will not work with an internal gear 
having more than 3.56 teeth. 

For unequal teeth we can use the formulae, 
2tt jyr 



tan. h = 



7l(JV+7l) + 4 7T* 



tan. H = 



2 7T n 



N{N-\-n)+±ir* 



in which N and n are the numbers of teeth 
in the pair of pointed gears. By these form- 
ulae the following results were determined, 
N K 



n 

2.695 
3. 
4. 
4.62 



10.17 
5.57 
4.62 




25° 27' 
33° 17 
34° 11 



71. — MINIMUM NUMBERS FOR UNSYMMETRICAL TEETH. 



If we drop the condition that the fronts 
and backs of the teeth shall be alike we have 
an unimportant case that is similar to that 
already studied, but much more intricate. 

If we carry this case to its extreme, and 
adopt single actiDg teeth, we have no mini- 
mum numbers at all, for any two numbers 
of teeth will then work together. Fig. 52 
shows one tooth working with three teeth, 
and any other combination can be obtained. 
The minimum obliquity for a given pair is 
obtained, as in (66), by laying off the known 
pitch arc, G D, at right angles to C e, and 
drawing the lioe of action at right angles 
to the line D c. The obliquity is also given 
by the formula : 

tan.K= -jt-j — , 

JV-\-n 

in which n and JY are the numbers of teeth. 
When the obliquity is as great as is often 



Fig. 52. 




Unsymmetrical 
teeth 



the case for very low numbers of teeth the 
action may be impracticable on account of 
the great friction of approach (48). The 
gears of Fig. 52 will not drive each other on 
the approach, unless the tooth surfaces are 
very smooth, and the power transmitted is 
almost nothing. 



72. — MINIMUM NUMBERS FOR GIVEN ARC OF RECESS. 



It has generally been assumed, although 
no good reason for the assumption has ever 
been given, that the minimum numbers of 
teeth occur when the tooth of one of the 
gears, Fig. 53, is pointed at the interference 
point I, and at the same time has passed 
over an arc of recess a that is a given part 
of the whole pitch arc a' a. 

The solution is simple enough, graphically 
by repeated trials, or by a formula that can 
be applied directly without the usual process 
by trial and error. 

But, as involute teeth have a uniform ob- 
liquity, there is no necessity for assuming 




Fig\ 53,, 

a definite arc of recess, and the condition on 



Involute Efficiency. 



37 



which the problem is based is unwarranted. 
No real limit is reached, and the matter is 
not worth examination at any length. The 
problem is investigated, for both bevel and 



spur gears, in either external or internal con- 
tact, in the Journal of the Franklin Institute 
for Feb., 1888, and it has received more atten- 
tion than its slight importance entitles it to. 



73. — EFFICIENCY OF INVOLUTE TEETH. 



But little can be said in addition to the 
matter in (49), for both forms of teeth in 
common use are substantially equal with re- 
spect to the transmission of power. 

From the formula of (49), which is the 
formula for the involute tooth, it is seen that 
the loss from friction is entirely independent 
of the obliquity, and, therefore, all systems of 
involute teeth are independent of the ob- 
liquity in this respect. This is contrary to 



the accepted idea that a great efficiency re- 
quires a small obliquity. 

It has been stated on high authority that 
the involute tooth is inferior to the cycloidal 
tooth in efficiency, but the statement is not 
true. The difference in efficiency is minute, 
a small fraction of one per centum, but what 
little difference there is is always in favor of 
the involute tooth. 



74. — OBLIQUITY AND PRESSURE. 



The involute tooth ^ction is in the direction 
of the line of action, and the obliquity is 
a constant angle. It is variable only when 
the shaft center distance is varied. 

As the pressure is always equal to the 
product of the tangential force at the pitch 
line multiplied by the secant of the obliquity, 



(26), it is constant for the involute tooth. 
Involute teeth, therefore, have a steady ac- 
tion that is not possessed by other forms ; 
particularly by forms which, like the cy- 
cloidal, have a pressure and an obliquity that 
varies between great extremes. 



70. — THE ROLLER OF THE INVOLUTE. 



The involute odontoid, like all possible 
odontoids, can be formed by a tracing point 
in a curve that is rolled on the pitch line, and 
this roller is the logarithmic spiral with the 
tracing point at its pole, (32). 

This feature is, however, more curious than 
useful, and it is not of the slightest im- 
portance in the stui3y of the curve. Neither 
is the operation of i oiling the involute me- 
chanically possible, fox the logarithmic roller 
has an infinite number Oi ^nvolutio as about 



its pole, and the tracing point would never 
reach the pitch line. 

The involute is often considered to be a 
rolled curve, because it can be formed by 
a tracing point in a straight line that rolls on 
its base line; but, although that is the fact, it 
is a special feature and has nothing to do 
with the rolled curve theory. The rolled 
curve theory requires that the odontoid shall 
be forme d by a roller *nat rolls on the pitch 
line only . 



A. THE CYCLOIDAL SYSTEM. 



76.— THE CYCLOIDAL SYSTEM. 



If the curve known as the cycloid is 
chosen as the determining rack odontoid, 
(31). the resulting tooth system will be 
cycloidal. 

It is commonly called the " epicycloidal" 
system, because the faces of its teeth are 
epicycloids, but, as the flanks are hypocy- 
cloids, it seems as if the name "epihypo- 
cycloidal " would be still more clumsy and 
accurate. 

There is no more need of two different 
kinds of tooth curves for gears of the same 



pitch than there is need of two different, 
kinds of threads for standard screws, or 
of two different kinds of coins of the same 
value, and the cycloidal tooth would never 
be missed if it was dropped altogether. But 
it was first in the field, is simple in theory, is 
easily drawn, has the recommendation of 
many well-meaning teachers, and holds its 
position by means of ''human inertia," or 
the natural reluctance of the average human 
mind to adopt a change, particularly a 
change for the better. 



77. — THE CYCLOIDAL TOOTH. 



The cycloid is the curve A that 
is traced by the point p in the circle 
G that is rolled on the straight pitch 
line p I, Fig. 54. The normal at 
the point p is the line p q to the 
point of tangency of the rolling 
circle and the pitch line. 

The line of action is the circle I a, 
of the same size as the roller G. 

As no tangent arc can be drawn to 
the line of action from the pitch 
point as a center, no terminal 
point (18) exists. As there is no 
point upon the line of centers from 
which a circle can be drawn tangent 
to the line of action, there will be no 
cusps, (16) except on the pitch line. 

The cycloidal tooth can be drawn 
by the general method of (24), but 
there are several easier methods 
which will be described. There 
are numerous empirical rules and 
short cuts to save labor and spoil 
the tooth, which will not be de- 
scribed. 

When the pitch line is of twice the diame- 
ter of the line of action, the flank of the 
tooth is a straight line. If the pitch line is 
less than twice as large as the line of action, 
the flank of the tooth will be under-curved, 




as shown by Fig. 55, and it is customary to 
avoid the resulting weak tooth by limiting 
the line of action to a diameter not greatei 
than half that of the smallest gear to be 
used. 



Cycloidal Secondary Action. 



39 



78. — SECONDARY ACTION. 



The secondary line of action (21) is 
a circle, Fig. 56, differing from tbe 
pitch circle by the diameter of the 
primary line of action, either inside or 
outside of it. 

When the internal secondary line of 
action of an internal pitch line coin- 
cides with the external secondary line 
of action of its pinion there will be 
secondary contact between the gears, 
the face of the gear working with the 
face of the pinion at a point of contact 
uyon the combmed secondaries. Fig. 
57 shows this for the cycloidal tooth, 
the two faces working together at the 
point a. As both secondaries are cir- 
cles they must coincide, and the sec- 
ondary action will be continuous. 

When the teeth are also in contact at 
b on the primary line of action, there 
will be double contact. 




V 

/ 
/ 



Undercurved flanks 

Fig, 55, 




Secondary lines of action 

Fig\ 56, 



40 



Cycloidal Interference. 



79.— INTERNAL INTERFERENCE. 



If the secondary lines of action do not 
come together the teeth will not touch each 
other at all, but if that of the gear is smaller 
than that of the pinion the teeth will cross 
each other and interfere. The line c, Fig. 
57, is the face of the gear tooth, and the line 
d is the face of the pinion tooth having a 
primary line of action equal to the difference 
between the pitch lines. The secondary line 
of each gear coincides with the pitch line of 
the other, and the faces interfere with each 
other the amount shown by the shaded 
space. 

The only remedy for internal interference 
is to reduce the diameter of the primary line 
of action to half the difference between the 
diameters of the pitch lines, or else to leave 
off one of the faces of the teeth. 

The discovery of the law of internal cycloid- 
al interference is due to A. K. Mansfield, 
who published it in the "Journal of the 
Franklin Institute" for January, 1877. It 
was afterwards re-discovered by Professor 
MacCord, and most thoroughly applied and 
illustrated in his " Kinematics." 

When interference is avoided by omitting 
one of the faces of the teeth the primary line 
of action may be enlarged, but it must not 
then be larger than the difference between 
the pitch diameters. 

Fig. 58 shows on the right the action 
when the face of the gear is omitted, and on 
the left the action when the face of the pin- 
ion is left off. The teeth will just clear each 
other, each one touching the other at a single 
point a in its pitch line. 

As the contact at a is not a point of practi- 
cal action, care must be taken that the arc of 
action at the primary line of action is as 
great as the circular pitch, for otherwise, as 
in the figure, the gears will not be in continu- 
ous primary action. 

The rule for internal interference, simply 
stated, is that the diameters of the pitch lines 
must differ by the sum of the diameters of 
the lines of action if the teeth have both 
faces and flanks, and by the diameter of the 
acting line of action if the face of either gear 
is omitted. For the standard interchangeable 
system the gears must differ by twelve teeth 




Secondary 
action 



Fig 




Internal 
interference 



I 



Fig. 58. 

if both teeth have faces, and by six teeth if 
one face is omitted. 

Fig. 62 shows the secondary contact in the 
case of a standard internal gear of twenty- 
four teeth working with a pinion of twelve 
teeth, and it is to be noticed that the teeth 
nearly coincide between the two points of 
contact. Where there is secondary contact 
the teeth practically bear on a considerable 
line instead of at a point. 



Cyclo /da I Odon tografi h . 



41 



80. — THE STANDARD TOOTH. 



The standard tooth (42), selected for the 
cycloid al system, is by common consent the 
one having a line of action of half the diame- 
ter of a gear of twelve teeth, so that that 
gear has radial flanks. 

The standard adopted by manufacturers of 
cycloidal gear cutters is that having radial 
flanks on the gear of fifteen teeth, but it is 
not and should not be in use for other pur- 



poses. If any change is made, it should be 
made in the other direction, to make the set 
take in gears of ten teeth. 

It must be borne in mind that the standard 
adopted does not limit the set to the stated 
minimum number of teeth, but that it sim- 
ply requires that smaller gears shall have 
weak under-curved teeth. 



81. — THE ROLLLD CURVE METHOD. 



It happens in this case, and in this case 
only, that the rolled curve method, which 
theoretically applies to all odontoids, can 
be actually put into practical use, for the 
generating roller is here the circle, the sim- 
plest possible curve. 

As in Fig. 59, roll a circle of the diameter 
of the circle of action upon the outside of the 
pitch line for the faces, and upon the inside 
for the flanks, and a fixed point in it will 
trace the curve. 

The method can be used by actually con- 
structing pitch and rolling circles, but the 
same result can be reached more easily and 
quite as accurately by drawing several cir- 
cles, and then stepping from the pitch point 
along the pitch line, and back on the circles 
to the desired point. If the length of the 




Construction by rolling 

Fig. 59. 

step is not more than one-tenth of the diam- 
eter of the circle, the error will not be over 
one-thousandth of an inch for each step. 

This method is the best one to adopt, ex- 
cept for the standard tooth. 



82. — THE THREE POINT ODONTOGRAPH. 



It is a simple matter to draw the tooth 
curve by means of rolling circles, but such 
a method requires skill on the part of the 
drafisman. It is, moreover, nothing but a 
method for finding points in the curve for 
which approximate circular arcs are then 
determined. 

The "three point" odontograph is sim- 
ply a record of the positions of the centers 
of the circles which approximate the most 
closely to the whole curve of the standard 
tooth. The positions of two points, a at the 
center of the face or of the flank, Fig. 60, 
and b at the addendum point or root point 
of the curve, were carefully computed, and 
then the position of the center C of the 
circle which passes through these two 



points and the pitch point 0, was calcu- 
lated. The circle that passes through these 
three points is assumed to be as accurately 
approximate to the true curve as any pos- 
sible circular arc can be. 

The odontograph gives the radius "rad." 
of the circular arc, and the distance "dis." 
of the circle of centers from the pitch line, 
for the tooth of a given pitch, and their 
values for other pitches are easily found by 
simple multiplication ur division. 

The advantages of this method lie in the 
facts that the desired radius and distance 
are given directly, without the labor of find- 
ing them, and that as they are computed 
they are free from errors of manipulation. 
In point of time required, the advantage is 



42 



Cycloidal Odontograph. 



with the odontograph in the ratio of 
ten to odc 

The greatest error of the odoDto- 
graphic arc, shown greatly exaggerated 
by the dotted lines, is at the point c on 
the face, and it is greater on a twelve- 
toothed pinion than on any larger gear. 
For a twelve-toothed pinion of three- 
inch circular pitch, a large tooth, the 
actual amount of the maximum error is 
less than one one-hundredth of an inch, 
and its average for eight equidistant 
points on the face is about four-thousandths 
of an inch. Any error that is greater than 



„ n f flank revters 
line ojj. - 




that stated will be due to manipulation, and 
not to the method. 



83. — USING THE ODONTOGRAPH. 



To apply the odontograph to any particu- 
lar case, tirst draw the pitch, addendum, 
root, and clearance lines, and space the piich 
line, Figs. 60 and 61. 

Then draw the lice of flank centers at 
the tabular distance "d is." outside of the 
pitch line, and the line of face centers at the 



distance "dis." inside of it. Take the face 
radius "rad." on the dividers, and draw in 
all the face curves from centers on the line 
of face centers; then take the flank radius 
"rad." and draw all the flank curves from 
centers on the liDe of flank centers. 



THREE POINT ODONTOGRAPH. 
Standard Cycloidal Teeth, 
interchangeable series. 
From a Pinion of Ten Teeth to a Rack. 





For One 


For One Inch 




For a 


DIAMETRAL PITCH. 


eby 


For a 


CIRCULAR PITCH. 


ply by 


NUMBER OP 


my other pitch divid 


ay other pitch multi 


TEETH 


that pitch. 


that pitch. 


IN THE GEAR. 








Faces. 


Flanks. 


Faces. 


Flanks. 


Exact. 


Intervals. 


Rad. 


Dis. 


Rad. 


Dis. 


Rad. 


Tis. 


Rad. 


Dis. 


10 


10 


1.99 


.02 


— 8.00 


4.00 


.62 


.01 


—2.55 


1.27 


11 


11 


2.00 


.04 


— 11.05 


6.50 


.63 


.01 


—3.34 


2 07 


12 


12 


2.01 


.06 


00 


oo 


.64 


.02 


CO 


oo 


13>4 


13—14 


2.04 


.07 


15.10 


9.43 


.65 


.02 


4.80 


3.00 


v^i 


15-16 


2.10 


.09 


7.86 


3.46 


.67 


.03 


2.50 


1.10 


m& 


17-18 


2.14 


.11 


6.13 


2.20 


.68 


.04 


1.95 


.70 


20 


19—21 


2.20 


.13 


5.12 


1.57 


.70 


.04 


1.63 


.50 


23 


22-24 


2.26 


.15 


4.50 


1.13 


.72 


.05 


1.43 


.36 


27 


25-29 


2.33 


.16 


4.10 


.96 


.74 


.05 


1 30 


.29 


33 


30-36 


2.40 


.19 


3.80 


.72 


.76 


.06 


1.20 


.23 


42 


37—48 


2.48 


.22 


3.52 


.63 


.79 


.07 


1.12 


.20 


58 


49—72 


2.60 


.25 


3.33 


.54 


.83 


.08 


1.06 


.17 


97 


73-144 


2.83 


.28 


3 14 


.44 


.90 


.0y 


1.00 


.14 


290 


145-300 


2.92 


.31 


3.00 


.38 


.93 


.10 


.95 


.12 


00 


Rack 


2.96 


.34 


2.96 


.34 


.94 


.11 


.94 


.11 



Cycloidal Odontograph. 



43 



The table gives the distances aDd radii if 
the pitch is either exactly one diametral or 
one inch circular, and for any other pitch 
multiply or divide as directed in the table. 

Fig, 61 shows the process applied to a 
practical case, with the distances given in 
figures. 



Fig. 62 shows the came process applied to 
an internal gear of twenty-four teeth work 
ing with a pinion of twelve teeth. It illus 
trates secondary action and double contact, 
It also shows the actual divergence of the 
Willis odontographic arc from the true 
curve. 



„ * flan h centers 




Internal teeth 

Fig. 02. 



44 



Willis Odo n tograpk . 



84. — THE WILLIS ODONTOGRAPHY 



This is the oldest and best known 
of all the odontographs, but it is 
inferior to several others since pro- 
posed, not only in ease of operation, 
but in accuracy of result. 

To apply it, find the pitch point 
a and a' half a tooth from the pitch 
point 0, Fig. 63, draw the radii a c 
and a' c', lay off the angles cab and 
d a' b\ both 75°, and Jay off the 
aistances a b and a' b' that are given 
by table. 

The centers b and b' thus found are 
The centers of circular arcs that are 
tangent to the tooth curves at d 
and d' . The dividers are set to the 
radius b or V to draw the curves. 

The Willis arc touches the true curve only 
at the pitch point 0, and its variation else- 
where is small, but noticeable. On the face 
of the tooth of a twelve-toothed pinion of 
three inch circular pitch, its error at the ad- 
dendum point is four-hundred i lis of an inch, 
and it will average three times that of the 
three point method (82). The error is shown 
by Fig. 62. 

The greatest error of the method is due to 
manipulation. The angle is usually laid off 
by a card, and the center measured in by a 
scale on the card. The circle of centers is 




The Willis odontograph 

Fig. 63. 



then drawn through the center, and unless 
great care is used the chances of error are 
great. 

The angle 90°^c ab = W =—, and the 

s 

distance ab = x— • sin. W, in which s 

2tt t ± s 

is the number of teeth in the gear of ihe 
same set which has radial flanks, usually 
12 ; c is the circular pitch, and t is the num- 
ber of teeth in the gear being drawn. The 
positive sign is used for the face raoius, and 
the negative for the flank radius. 



85. — KLEIN'S CO-ORDINATE ODONTOGRAPH. 

This is a method of finding the positions 
of several points on the tooth curve by 
means of their co-ordinates referred to axes 
through the pitch point. Any point on the 
curve is found by laying off a certain dis- 
tance on the radius Y, Fig. 64, and then 
a certain distance at right angles to it, the 
distances being given by a table for a certain 
standard tooih. 

As many points as rtquired are found by 
this method, and then the curve is drawn in 
by curved rulers, or by rinding the approxi- 
mating circular arc. 

This odontoorraph is to be found in Klein's 
Elements of Machine Design. 




Coordinate odontoyraph 

Fig. 64. 



Obliquity of Action. 



45 



86. — THE TEMPLET ODONTOGRAPH. 



Prof. Robinson's templet odontograph is 
an instrument, not a method. It is a piece 
of sheet metal, Fig. 65, having two edges 
shaped to logarithmic spirals. It is laid 
upon the drawing, according to directions 
given in an accompanying pamphlet, and 
used as a ruler to guide the pen. It can be 
fastened to a radius bar, and swung on the 
center of the gear, to draw all the teeth. 
See Van Nostrand's Science Series, No. 24, 
for the theory of the instrument in detail. 




The iemplet odontograph 

Fig. 65. 



87. — OBLIQUITY OP THE ACTION. 



When the point of contact between two 
teeth is at the pitch point 0, Fig. 66, the 
pressure between the teeth is at right angles 
to the line of centers, but, as the point of con- 
tact recedes from the line, the direction of 
the pressure varies by an angle of obliquity 
which increases from zero until the point K, 
at the intersection of the addendum circle 
with the line of action, is reached. 

The angle K — K W, of the maximum 
obliquity, can be found by solving the trian- 
gle G c K, and for the standard set we have, 

cos. 2 K = ^ i—r^, 

3 n-f- 18 

in which n is the number of teeth in the 

gear. 

For the smallest gear of the set, the one 
having twelve teeth, K is 20° 15', and for the 
rack it is 24° 5', so that it will always be be- 
tween those two limits for external gears, 
and greater for internal gears. 

The friction between two gear teeth in- 
creases with the angle of obliquity, but not 




Obliquity 

Fig. 66. 

in direct proportion. With the involute 
tooth the work done while going over a cer- 
tain arc from the line of centers is propor- 
tional to the square of the arc, and for 
cycloidal teeih the increase with the arc is 
still more rapid. Therefore it is the maxi- 
mum obliquity of the action that principally 
determines the injurious effects of friction. 



— THE CUTTER LIMIT, 



When the number of teeth in the gear is 
less than that in the gear havin g teeth with 
radial flanks, the flanks will be under-curved, 
and when too much so they cannot be cut 
with a rotary cutter. Tbe teelh of Fig. 55 
could not be cut with a rotary cutter beyond 
the points where the tangents to the two 
sides are parallel. 

The limit is reached when the last point 



that is cut by the rotary cutter is also the 
last point that is touched by the tooth of the 
rack in action with it, not allowing for in- 
ternal gears. 

Toe diameter of the gear when this limit 
is reached is found by the formula, 



D = 2ff — 



2 sin. 



r a 



46 



Limiting Cycloidal Teeth. 



in which D is the diameter of the gear, d is 
the diameter of the circle of action, c is the 
circular pitch, and a is the addendum 

For the common addendum of unity- 
divided by the diametral pitch this may be 
put in the shape, 

n — s 



2 sin. 



V 



in which s is the number of teeth in the 
radial flanked gear, and n is \q number in 
the required cutter limn. 

For the common series, where * = 12, we 
have n = 8.26; and for the cutter standard of 
s = 15, we have n = 10.80, so that cutters 
could easily be made to cut gears with less 
than s teeth. 



89. — RADIAL FLANKED TEETH 

When the rolling circle for 
the faces is of half the diam- 
eter of the pitch line of the 
mating gear, the flanks of 
both gears will be straight 
radial lines, as in Fig. 67. 

Such gears are fitted to each 
other in pairs, and are not 
interchangeable with other 
sizes. Their teeth are more 
easily made than those of 
standard gears. The maxi- 
mum obliquity is less, but 
the strength of the teeth is 
also less than usual. There 
is no reason for making such 
teeth in preference to the 
though, for that reason probably, they are 
used to a considerable extent. It would be 




Itadial flanks 

Fig. 67. 



standard, al- 



difficult to devise a form of tooth so whimsi- 
cal that it would find no one to adopt and 
use it. 



90. — THE LIMITING NUMBERS OF TEETH. 



When the number of teeth in a driving 
gear is small, the point p, Fig. 68, of its 
pointed tooth may go out of action by leav- 
ing the line of action g before a certain 
definite arc of recess r has been passed over, 
and the problem is to find the smallest num- 
ber of teeth in the following gear that will 
just allow the given recess. 

This question, which is not a particularly 
important one, is discussed at length, and 
applied to both bevel and spur gears, in either 
external or internal contact, in an article in 
the "Journal of the Franklin Institute" for 
Feb., 1888, and we will here consider only 
'he case of the common spur gear. 

The recess r is given as a times the cir- 
cular pitch, and the thickness a r of the 
tooth is given as b times the same. The 
diameter of the circle of action is q times 




Limiting tooth 

Fig. 68. 



that of the pitch line of the following gear. 
The number of teeth in the driving gear is d, 
and the number in the following gear is /. 



The Pin Tooth. 



360 a 



and 



M is an auxiliary angle equal to 

ttt. i 360/ b\ 

W is an angle — —[& — j. 

Then the required number / can be found 
by a process of trial and error with the 
formula, 



sin. (M-\- W) d 



sin W 



_- -1 = 0. 



For an example, let the recess be £ of the 
pitch, the tooth equal to the space, and the 
flanks of the follower to be radial. Let the 
problem be to find-a follower for a driver of 
seven teeth. This gives # = £, b = i, q = ±, 
d = 7, and the formula becomes 

(™T + 25" 48' ) 



sin. 



sin. 25° 43' 



f 



— 1 = 0. 



If we put / at random, at 20, we snail 
get, -f- .134 = 0. Next, trying /= 10, we 
get, — .132 = 0, and the opposite signs show 
that / is between 20 and 10. Trying 12 the 
result is positive, and for 11 it is negative, 
showing that 12 is the required value of /. 
That is, 7 teeth will not drive less than 12 
teeth with radial flanks, unless ic is allowed 
an arc of recess greater than £ of the pitch. 

For another example, test MacCord's value 
of 382 as the least driver for a follower of 
10 teeth, when rtcess equals the pitch and 
the follower has radial flanks. Trying d = 
382, the error is negative ; for 383 it is also 
negative, but for 384 it is positive, and there- 
fore the latter is the true number. 

Extensive and sufficiently accurate tables of 
limiting values are given by MacCord in his 
" Kinematics." 



5. THE PIN TOOTH SYSTEM. 



91. — THE PIN GEAR TOOTH. 



The theory of the piD gear tooth is en- 
tirely beyond the reach of the " rolled curve" 
method of treatment, and, therefore, writers 
who have adopted that method have had to 
depend more on special methods adapted to 
it alone than on general principles. The re- 
sult is that its properties are often given in- 
correctly, or with an obscurity and complica- 
tion that is bewildering to the student. 
Although the tooth is one of the oldest in 
uae, its theory is so difficult that its defect 
was not discovered until within a very few 
years, by MacCord, about 1880, and it was 



not until it was examined by means of its 
normals that a remedy for that defect was 
discovered. 

By treating the curve on the general prin- 
ciples here adopted, as a special form of the 
segmental tooth, it can be studied with ease, 
and its peculiarities developed in a complete 
and satisfactory manner. The method, in 
general terms, is to find the conjugate tooth 
curve of the gear, for the given circular tooth 
curve of the pinion, and it presents no new 
features or difficulties. 



92. — APPROXIMATE FORM OF PIN TOOTH CURVE. 



Considered roughly, but accurate enough 
for teeth of small size, the form of the 
gear tooth b, Fig. 69, is a simple parallel 
to the epicycloid E, formed by the center e 



any convenient number of circles having 
centers on the epicycloid. 

The action is practically all on one side of 
the line of centers, the face of the gear tooth 



of the pin, and is to be drawn tangent to working with the part of the pin that is 



48 



The Pin Tooth. 



inside of its pitch hue. It is, theiefore, all 
approaching action when the pin drives and 
all receding action when the gear drives, and 
it is best to avoid the increased friction of 
the approaching action by alwajs putting the 
pins on the follower. 




Lantern tvJieel 

Fig. 70. 




Pin gearing 

Fig. 60. 



93. — ROLLER TEETH. 



The pin gear is particularly valuable when 
the pins can be made in the form of rollers, 
Fig. 70, for then the minimum of friction is 
reached. The roller runs freely on a fixed 
stud, or on bearings at each end, and can be 
easily lubricated. 

The friction between the tooth and pin, 
otherwise a sliding friction at a line bearing, 
is, with the roller pin, a slight rolling fric- 
tion, and the sliding friction is confined to 



the surface between the roller and its bear- 
ings. 

When the roller pin is used there can be 
no increased friction of approach, and the 
pin wheel can drive as well as follow. 

For very light machinery, such as clock 
work, there is no form of tooth that is su- 
perior to the roller pin tooth, and, with the 
improvement to be explained, there is no 
better form for any purpose. 



94. — CUTTING THE PIN TOOTH. 

The pin gear tooth can be very easily and 
accurately shaped by mounting a revolving 
milling cutter M, Fig. 71, of the size of the 
pin, upon a wheel A, and causing it to roll 
with a wheel B, carrying the gear blank O. 
The mill will shape the teeth to the correct 
form. 




Pin gear cutter 

Fig. 71. 



95. — PARTICULAR FORMS OF PIN GEARS. 



When the pins are supported between 
two plates, as in Fig. 70, the wheel is called 
a "lantern" wheel, and is the most common 



form of clock pinion, 
times called " staves,' 
known as " leaves." 



The pins are some- 
and are sometimes 



Defect of Pin Tooth, 



49 



When the diameter of ihe pin is 
zero, Fig. 72, it being merely a 
point, the correct tooth curve will 
be a simple epicycloid. 

When the pin gear is a rack, Fig. 
73, the tooth bears on the pin only 
at a single point on the pilch line, 
and the action is therefore very oe- 
fective unless the roller form of pin 
is used. This form is more properly 
a particular case of the involute tooth, 
for the shape of the pin is immaterial i f it does 
not interfere with the gear tooth. The circle 
wiih center on* a straight line is not an 
odontoid at all, for, although it coincides as 
a whole and for a single instant with a cir- 
cular space in the gear, it has no proper and 
continuous tooth action. 

The gears of Fig. 74, sometimes classed 
with pin gearing, are not pin gears at all. 
An epicycloidal face working with a radial 
flank is a very common combination. 

•When the diameter of the pin wheel is half 
that of the internal gear with which it works, 
we have the combination of Fig. 75. The 
pins may run in blocks fitted to the straight 
slots. 




Point gears 

Fig. 72. 



Pin, rack 

Fig. 73. 





Jfot.pin gears 

Fig. 74* 



Radial 2) in teeth 

Fig. 75. 



96. — CORRECT FORM AND DEFECT OF PIN TEETH. 



Although the pin tooth is apparently of a 
very simple form, a close examination will 
show that it is really quite complicated, and 
that its practical action is incomplete and de- 
fective. There is a cusp (16), and conse- 
quent failure in the action, that is of small 
importance when the teeth are small, but 
which is troublesome when they are large. 
This defect need not be considered when 
pinions for clock work are in view, but if 
pin wheels are to be used for large machinery 
and heavy power it is important. 

If the pin a, Fig. 76, is examined as an 
odontoid, it will be seen that it is a true 
odontoid only within the line TeT that is 
tangent to the pitch line at the center of the 
pin, for all normals, as pe, from points out- 
side of that line, intersect the pitch line at 
the center. 

Drawing the normals, which are radii of 
the pin, we can easily construct the line of 



action and the conjugate tooth curve. The 
line of action, commencing at the pitch point 
0, Fig. 77, is there tangent to the line eOm, 
which passes through the center e of the pin, 
curves toward Oh, the tangent to the pitch 
line at the pitch point, and touches it at the 
point h, at the distance Oh, equal to the ra- 
dius of the pin. From the point h it follows 
the circle hjh' to the point h' . thence return- 
ing to the pitch point and forming the loop 
OKL. 

From the center c of the gear, Fig. 78, we 
can always draw a tangent arc FN to the 
line of action at the point F, and therefore 
there will always be a cusp at N on the 
tooth curve. The tooth curve must end at 
the cusp, and, to avoid interference, the pin 
must be cut off at the arc W, drawn through 
the point F, from the center C. 

The whole pin is generally used, and 
when it is a roller it must be whole, and 



50 



Improved Pin Tooth. 



then interference can be avoided only by 
cutting away the tooth curve until it will al- 
low it to pass. 

The complete tooth curve has a first 
branch NOM, Fip\ 78, which is the only 
part that can be used, an inoperative second 
branch from the first cusp N to the second 
cusp Q on the arc EQ, and thence an inopera- 
tive circle OBQ'. 




Z.ine of action 

Fig, 77. 




Correct action 

Fig, 78. 



97. — AN IMPROVED PIN TOOTH. 



The cause of the broken action of the pin 
tooth is the cusp, which is always present 
when the center of the pin is on the pitch 
line, and it can be avoided by placing the 
center back, as in Fig. 79, to such a distance 
inside the pitch line that the cusp does not 
occur. 

When the center of the pin is inside the 
pitch line, the whole circle of the pin is a true 
odontoid, and the distance en of the center 
from the pitch line can be so chosen that 
the cusp is not formed. 

This distance does not appear to be sub- 
ject to any simply stated rule, but in the 
single case of the pin rack it is determined by 
the formula: 

2 (? a 

" 27 I) ' 

in which x is the required distance en, D is 
the diameter of the gear, and d is the diame- 
ter of the pin. 



If the angle CeO, Fig. 79, is not less than 
a right angle, there will be no cusp on the 



Flg r 70. 




Corrected pin gear 



gear tooth if the diameter of the gear is 
greater than that of the pin. 



6. TWISTED, SPIRAL, AMD WORM GEARS, 



98. — STEPPED GEARS. 



When two or more gears, Fig. 80, of the 
same pitch diameter, are placed in contact en 
the same shaft, they will evidently act as in- 
dependently of each other as if they were 
some distance apart, while they appear to act 
together as a single gear with irregular teeth. 
They are known as " Hooke's Gears." 

It matters not. how many different kinds or 
numbers of teeth the several gears may have, 
or in what order they are arranged, if those 
that work together on opposite shafts are 
matched. They may be given an irregular 
arrangement, as in Fig. 80 ; a spiral arrange- 
ment, as in Fig. 81 ; a double spiral, or "her- 
ring-bone " arrangement, as in Fig. 82 ; a cir- 
cular arrangement, as in Fig. 83, or other- 
wise at will. 





Fig. 80. 



Stepped year 




Spiral arrangement 

Fig. 81. 




Double spiral arrangement 

Fig. 82. 



Circular arrangement 

Fig. 83. 



99. — TWISTED TEETH. 



The thickness of the component gears has 
nothing to do with the theoretical action of 
the stepped gear as a whole, and therefore 
we can have them as thin as required. If the 
thickness is infinitesimal the component 
character of the gear is not apparent, and it 
is known as a twisted gear, Fig. 84. 

When the teeth are twisted there may al- 
ways be one or more points of contact at the 
line of centers, where the theoretical fric- 
tion is nothing, and therefore they are par- 
ticularly well suited for rough cast teeth. 
Furthermore, if the teeth are badly shaped 




Twisted arrangement 

Fig. 84. 



52 



Twisted Teeth. 



the twisted arrangement tends to distribute 
the errors so that they are not as noticeable. 
The oblique action of twisted teeth tends 
to produce a longitudinal motion of the 
gears upon their shafts, which must be 
guarded against. This end thrust may be 
avoided by so forming the twist that there 



are aiways two oblique bearings between the 
teeth, acting in opposite directions, as in the 
herring-bone arrangement. 

The twisted form of tooth is seldom found 
in practice, except in the form of spiral and 
double spiral teeth, for the difficulty of form- 
ing other twists is great. 



100. — EDGE TEETH. 



If the twist of the twisted tooth is such 
that some part of the twist at the pitch cylin- 
der is always upon the line of centers, the 
gears will always be in action whether 
there are full teeth or not, and they will 
work with theoretical accuracy if they are 
reduced to edges in the pitch cylinder, as in 
Fig. 85. 

The friction of the edge tooth is theoret- 
ically nothing, as there is no sliding of the 
teeth on each other. There is but one point 
of contact, and that is always upon the line 
of centers; but if any power is carried the 
pressure will soon destroy the single point of 
contact. 




Kdye teeth 



If the edges are thick the action will be 
stronger, but there will still be but one point 
of contact. 



101. — INVOLUTE TWISTED TEETH. 

When the form of the tooth is the invo- 



lute, and the twist is such that some part of 
it on the pitch cylinder always crosses the 
line of centers, the teeth will remain in con- 
tact, when the parallel axes are separated, 
until their points are separated, although the 



contact may sometimes be very short or even is variable 



point contact. The straight involute tooth 
will fail as soon as the arc of contact is less 
than the tooth arc. 

Twisted involute, teeth are therefore partic- 
ularly valuable for gears for driving rolls, or 
for other purposes where the shaft distance 



102. — FORMATION OF THE TWISTED TOOTH. 



When the twist is a uniform spiral there 
are convenient methods for shaping the 
tooth, but the twisted tooth in general can be 
formed only by the processes of (27), (28) and 
(29), and then only when the twist is not 
very irregular. 

The principle of the linear planing opera- 



tion of (29) is the same as for the straight 
tooth, but the blank must be rotated accord- 
ing to the form of the twist adopted, while 
the tool is cutting. The twisting motions 
are independent of the feeding motion, and 
are repeated at every stroke. 



103. — SPIRAL GEARS. 



The spiral gear is that particular form of 
the twisted gear which has uniformly 
twisted teeth, and it is, therefore, a particu- 



lar form of the common spur gear. It has 
such peculiar properties that it is often 
classed by itself as a separate form of tooth. 



Spiral Act i 07i. 



The normal spiral section is that section of 
the teeth of the spiral gear that is made by a 
spiral surface, called a helix, that is at right 
angles with the teeth. It is the equivalent, 
for spiral teeth, of the normal section of the 
spur gear that is made by a plane, or of the 
normal section of the bevel gear that is made 
by a sphere. As with spur and bevel gears, 
the action of the teeth on each other should 
be studied upon this normal surface. As the 
helix cannot be represented upon a plane 
figure it must be imagined, and as it is ob- 
scure it requires close attention. 

Any two spiral teeih will work together, 
provided their normal spiral sections are con- 
jugate (24), and, as the shape of the normal 
spiral section is independent cf the angle of 



the spiral, two spiral gears will work to- 
gether, approximately, on shafts that are 
askew. This will be seen more clearly if the 
spiral section is imagined to be a flexible 
sheet-metal toothed helix, which can be coiled 
about the shaft of the gear, for it can evi- 
dently be coiled close or loose without affect- 
ing the shape of its teeth. If coiled clo*e, 
with a short lead, it runs nearly at right 
angles to the shaft, and the gear approxi- 
mates to the spur gear, while if the lead is 
long the gear approximates to the screw. 

As the diameter of the spiral gear increases, 
the teeth straighten, and when the diameter 
is infinite and it is a rack, they are straight 
and in no way different from those of a com- 
mon rack. 



104. — THEORY OF SPIRAL TOOTH ACTION. 



The "Willis theory of the action of spiral 
teeth is the one generally accepted, but it is 
not correct. It assumes that the action be- 
tween the gears is upon a section by a plane 
through the axis of the gear and the common 
normal to the two axes, and that the section 
of the two gears made by the plane act to- 
gether like a rack and gear. 

When the axes are at right angles, and the 
spiral angle is great, this theory is apparently 
correct, the error being practically imper- 
ceptible, but, as the axes become more nearly 
parallel, the error is more apparent, until, 
when they are parallel, the error is plain 
enough. Willis applied his theory to worms 
and worm gear?, on axes at right angles, and 
evidently did not consider the spiral gear in 
general. 

The action between spiral teeth is not upon 
the axial section, and it is not that of a rack 
and gear, but when there is any action at all 
it is upon the normal spiral section. See the 
American Machinist for May 19th, 1888. 

When the axes are parallel the normal 
spiral sections, as well as the sections made 
by a plane normal to the axes, are conjugate, 
and therefore the action is correct and along 
a line of action The action is also continu- 
ous when the axes intersect and the gears are 
bevel gears. 



When, however, the axes are askew, the 
normal spiral sections are not necessarily con- 
jugate, for they coincide only on one line, the 
common normal to the two axes. Therefore, 
there is no continuous tooth contact, except 
in one particular case, the teeth being in con- 
tact only for an instant as they pass the 
normal. 

The special case for which spiral teeth on 
askew axes have continuous tooth contact, is 
that case of the involute tooth when the base 
cylinders are tangent and the gears become 
spiraloidal skew bevel gears. See (175) and 
(176). In that particular case the teeth have a 
sliding conjugate action on each other. As 
the spiraloidal gear is fully described in its 
place, it will not be further considered here. 

This theory is corroborated by experiment- 
al gears made for the Brown & Sharpe Man- 
ufacturing Company, for whom Mr. O. J. 
Beale, to whom the theory of the spiral 
gear is much indebted, made a pair of theo- 
retically perfect spiral gears, exactly alike, 
with a spiral angle of 45°, working on shafts 
at right angles, and of such a large size that 
the action of the teeth could be plainly ob- 
served. See Figs. 88 and 90. 

Beale's gears cannot be made to run to- 
gether properly at any shaft distance, but 
if i heir ends are brought to the common 



54 



Spiral Gearing. 



normal, and their base cylinders are in con- 
tact, they are skew bevel gears and show the 
action required by Olivier's theory. 

But, although the action of spiral gear 
teeth is intermittent, and their contact is the- 
oretically perfect at one instant only, when 



they are passing the common normal, they 
are very nearly in contact all the time and 
the action is practically perfect. Spiral teeth 
of ordinary sizes work together with a re- 
markably smooth action. 



105. — FORMATION OF THE SPIRAL TOOTH. 



As the spiral rack has an ordinary straight 
tooth, we can conveniently derive the spiral 
tooth in general from it by a method that is a 
form of the molding method of (27) for spur 
gears. 

If a plane is moved in any direction upon a 
cylinder it will move it, as if by friction, 
with a speed that depends upon the direction 
of the motion. If we imagine the same re- 
sulting motion between the plane and the 
pitch cylinder, and assume that the plane is 
provided with hard and straight teeth run- 
ning in any direction, it will mold the plastic 
substance of the cylinder and form spiral 
teeth upon it. All spiral teeth formed by the 
same rack will have normal spiral sections 
that are approximately conjugate to each 
other, and they will work together inter- 
changeably. 

This process may be put into practical 
shape by a modification of the process of (28) 
for spur gears, by substituting a planing 
tooth for the molding rack tooth. The tooth 
has the shape of the normal section of the 
rack, and, as it is reciprocated at an angle 
with the axis of the gear blank being shaped, 
both the tool and the gear blank receive the 
motion of the plane and pitch cylinder. The 
cutting face of the tool is normal to the direc- 
tion of its motion, which motion is tangent 
to the direction of the tooth spiral. 

The linear process of (29) may be used, the 
plane of Fig. 20 representing, approximately, 
the normal spiral section of the gear. Thus, 
if the planing tool or the equivalent milling 
cutter receives a motion as if in a plane roll- 
ing upon the base cylinder, the involute tooth 
will be produced. 



The spiral tooth may be formed by the 
linear planing process of (29), directly ap- 
plied on the principle that the spiral tooth is 
a twisted spur tooth. The planing tool re- 
ceives a planing motion in the direction of 
the axis of the gear blank, and both tool and 
blank receive the feeding rolling motion that 
would produce the spur tooth of the section 
that is normal to the axis. In addition, the 
blank receives a motion of rotation while the 
tool moves, that is repeated for every troke 
of the tool. The cutting edge of the tool is 
set normal to the axis of the gear. 

The spiral tooth may also be formed bv a 
tool that is formed to the true shape of some 
section of the tooth, preferably its normal sec- 
tion, and which is guided in the tooth spiral. 
This is the process used to shape a worm, the 
tool being guided by a screw-cutting lathe. 

The process generally used to mill the 
teeth of the spiral gear is the equivalent of 
the operation last described. The milling 
cutter is shaped to the normal section of 
the tooth space, and is guided in the tooth 
spiral by a special feeding device that ro- 
tates the blank while the cutter is working 
in it. 

Of these processes the planing process of 
(28) is the best, as it produces the tooth with 
theoretical perfection, and because all gears 
formed with the same tool are conjugate and 
interchangeable. But the screw-cutting and 
milling processes are most in use, for the 
reason that they are more expeditious and 
better adapted to the common machine tools, 
and it is therefore necessary to study the 
shape of the normal section of the tooth with 
some care. 



Spiral Gearing. 



DO 



103. — THE NORMAL PITCH. 



The real pitch of the spiral gear is meas- 
ured on a section that is normal to its axis, 
and, as in the case of the spur gear, it is 
found by dividing the number of teeth by 
the pitch diameter, but the shape of the tooth 
must be regulated by the normal pitch, or 
pitch of its normal section. 

The normal pitch is found by dividing the 



real pitch by the cosine of the angle made by 
the tooth spiral with the axis of the gear. 
Thus, if the pitch is 8, and the angle is 45°, 
the normal pitch is 8, divided by ."<07, or 
11.3. 

The normal circular pitch is found by mul- 
tiplying the real circular pitch by the cosine 
of the spiral angle. 



107. — THE ADDENDUM. 

The addendum of the spiral gear should 
not be determined by its real pitch, but by 
its normal pitch, for it is then usually possi- 
ble to mill the tooth with a milling cutter 
that is made for a standard spur gear. A. 
gear of 8 pitch and 45° angle should have an 

addendum of ^ps = .089". 

If the addendum is determined by the true 
pitch when the angle is considerable, the 
tooth will be long and thin. Fig. 86 shows 
the normal pitch section of a rack to run 
with a pinion of 45° angle, while Fig. 87 
shows the true pitch of the same rack. Fig. 
88 also shows the true pitch of the pinion, 
and, although the tooth appears to be stunted, 
it is really of the standard shape. 




Formal Rack. 
Fig. 86. 




Spiral Rack. 
Fig, 87. 




Beale>8 Experimental Gears. 
Fig. 88. 



108. — THE AXTAL PITCH. 

The section of the spiral gear by a plane ' spiral angle. Thus, if the angle is 45°, the 
through the axis is that of a rack, and the ; axial pitch is the same as the true pitch, but 
axial pitch, or pitch of the rack, is found by the axial pitch of a 70° spiral tooth is but .364 
dividing the true pitch by the tangent of the | of the true pitch. 



109. — SHAPING THE TOOL. 



When the spiral gear is cut in a milling 
machine, or turned in a lathe, it is necessary 
to give the tool the shape of the normal sec- 
tion of the tooth to be cut, and this is most 
readily accomplished by shaping it for the 
spur gear that most nearly coincides with that 
normal section. 



The number of teeth in the gear that is 
osculatory to the normal spiral, and therefore 
most nearly coincides with it, is found by 
dividing the actual number of teeth in the 
gear by the third power of the cosine of the 
spiral angle. 

For example, if we are to cut a gear of 4" 



56 



Spiral Gearing. 



diameter, 6 pitch, and 24 teeth, at a spiral 

angle of 45°, the cutter should be shaped to 

„ 24 24 

cut a spur gear of 



of - ^ = 8.5 pitch. 



707* .35 = 69teeth 

If the gear has 28 teeth 



of 4 pitch, and an angle of 10°, the equiva- 
lent spur gear has 29 teeth of 4.08 pitch, as 
the gear varies but little from a spur gear. 
If the gear is of 5 pitch, and 15 teeth, with 
an angle of 80°, the equivalent spur gear has 
2,830 teeth of 28.7 pitch, and in general, 
when the gear has a great angle it is a 



worm, the section is practically that of a rack. 
Care must be taken, when the gear is a 
screw, and is turned in the lathe, that the 
tool should be set with its cutting edge nor- 
mal to the thread of the screw, if it is shaped 
by the above rule. If the tool is set in the 
axial section of the screw, and it generally is, 
it should be shaped to the axial section of the 
worm, and have the axial pitch and adden- 
dum. But when the lead of the thread of 
the screw is small compared with its diam- 
eter the difference between the normal and 
axial sections is not noticeable. 



110. — VELOCITY RATIO OF SPIRAL GEAIIS. 



The spiral gear does not follow the well- 
known rule of spur gears, that the velocities 
in revolutions in a given time are inversely 
proportional to the pitch diameters, but re- 
quires that ratio to be multiplied by the ratio 
of the cosines of the spiral angles. 

In the formula 

D cos. A 

= a 



V 

Y 



cos. a 

D and d are the diameters of the gears, A 
and a are their spiral angles, and V and v 
are their velocities in revolutions. 

If the angles are equal, the velocity ratio 
is the same as for spur gears of the same 
diameters. Fig. 88 shows a pair of gears B 
and G that are of the same size and have the 
same angle in opposite directions, requiring 
the shafts to be parallel. See also Fig. 89. 
The pair of gears A and B are exactly alike, 
with equal angles in the same direction, re- 
quiring the shafts to be at an angle equal to 




Fig. 89 



Spiral Sptiv Gears 




Equal 
Gears 



Fig. 90 



twice the spiral angle. See also Fig. 90. 
The statement that like spiral gears will not 
run together is founded on the Willis theory 
of spiral gear contact, and is wrong. 



111. — SPIRAL WORM AND GEAR. 



When the shafts are at right angles, and 
the angle on one is so great that it is a screw, 
the combination is known as a worm gear 
and worm, Figs. 91 and 92, and is much used 
for obtaining slow and powerful motions. 
It is also too much used for wasting power 
and wearing itself out, for its friction is very 
great and consumes from one-quarter to 
two-thirds of the power received . 

When the screw has a single thread, the 



velocity ratio is simply the number of teeth 
in the gear, and if there are two or three 
threads it must be modified accordingly. 

The spiral worm is adjustable in iis gear 
both laterally and longitudinally, so that it 
will change its position as required by wear 
in the shaft bearings. 

It is an excellent substitute for the hobbed 
worm and gear, and in most cases will serve 
practical purposes quite as well. 



Worm Gearing. 



;>/ 




Spfral Worm Gear and Worm. 
Fig. 91. 



Fig. 92 




Worm Gears 



112. — EFFICIENCY OF SPIRAL AND WORM GEARING. 



Unless the shafts are parallel the teeth of a 
pair of spiral gears are moving in different 
directions, and therefore they cannot pass 
each other without sliding on each other an 
amount that increases rapidly with the angle 
of divergence of the directions of motion, 
that is, the shaft angle. 

This sliding action creates friction and 
tends to wear the teeth, and to a very much 
greater extent than is generally supposed. 
The friction is so great, in fact, that such 
gears, particularly worm gears, should be 
used only for conveying light powers. They 
are extensively used, or rather misused, for 
driving elevators, and are even found in mill- 
ing machines, gear cutters, planers, and 
similar places, in evident ignorance that they 
waste from a quarter to two-thirds of the 
power received. 

The most extensive experiments on the 
efficiency of spiral and worm gears ever made 
were made by Wm. Sellers & Co., and they 
may be found described in great detail in a 
paper by Wilfred Lewis in the Transactions 
of the American Society of Mechanical En- 
gineers, vol. vii. Space will not permit ex- 
tensive quotations from this valuable paper, 
but the general result of the experiments is 



shown by the diagram, Fig. 93. The diagram 
shows that a common cast-iron spur gear and 
pinion on parallel shafts have an efficiency of 
from ninety to ninety-nine per cent., accord- 
ing to the speed at which they are working ; 
that a spiral pinion of 45°, angle working in 
a spur gear, with shafts at 45°, has an effi- 
ciency of from 81 to 97 per cent. ; that 
the efficiency decreases as the angle of the 
shafts increases, until, for a worm of a spiral 
angle of 5°, at a shaft angle of 85°, it goes 
as low as 34, and does not rise higher than 77 
per cent. This includes the waste of povrer 
at the shaft bearings as well as that at die 
teeth of the gears. The efficiency is lowest 
for slow speeds, and rises with the speed. 
The diagram may be relied upon to give its 
true value, under ordinary conditions, within 
five per cent. 

The same experiments developed the fact 
that the velocity of the sliding motion of the 
cast-iron teeth on each other should not be 
over two hundred feet per minute in contin- 
uous service, to avoid cutting of the surfaces. 
It may be assumed that the efficiency will be 
higher when the worm is of steel, particu- 
larly when the gear is of bronze. 

Diagram, Fig. 94, shows the result of simi- 



58 



Worm Gearing. 



Velocity at PitcJi X,ine in feet per minute. 



0« "O p 



C3 ^ »0 :0 t- x 3:^ — < .-. 



o 2°o oooo 



o o © o 



TDOliH « f-< 



m z3E 







































1 Spur Pinian 






































i ■ — y-fp — ; 7Z°~T~ ', 












































































... _ _ 








































-S}.. Pn, 30 








































hSjh Pn. 20" 


























































































































































































































































































































































































































































































































































































































•*a 








































































































































































































05 
















































































V,. 








































°-i 
















































































fe: 








































ii 








































g 
















































































































































































































































&~ 














































































h. 








































































































































































































































































































































































— 










































































































































































































































































































































































Si 












































































-r^ 




' 

































.95 



.90 



.85 



.SO 



75 



.40 



o oo ooooo o 

r-i —. —. <M 



Sellers 9 Experiments 

Fig. 93. 



lar experiments by Prof. Thurston, with a 
worm of 6" diameter and one inch circular 
pitch running in a gear of 16" diameter, both 
cast-iron. 

It is to be observed that it is the shaft angle, 
and not the angle of the spiral, that deter- 
mines the efficiency. A pair of spiral gears 
on parallel shafts are practically as efficient 
as gears with straight teeth. 

The great friction of worm gearing is of 
advantage for one purpose, and for one only, 
to secure safety and prevent undesired mo- 
tion of the gears. The worm of Fig. 97 will 
easily move the gear, but the gear must be 
moved with great force to start the worm. 
When the angle of the worm is as small as 
the "angle of repose" for the metals in 
contact, it is impossible for the gear to drive 
the worm. This may be an excuse for the use 
of the worm gear in elevators, but it would 
seem that the safety of the cage should de- 



Revolutions of Driver per minute 

50 100 150 200 250 300 350 400 




Yale & Toivne Experiments 

Fig. 04. 



pend on devices attached to the cage itself, 
rather than to the hoisting machinery or 
other distant part. 

Unless the friction of the gears must be 
depended upon for safety, the worm gear 
should be used only for purposes of adjust- 
ment, or when speed must be greatly reduced 
or power increased within a small compass, 
and not for conveying power. 



Worm Gearing'. 



59 



113. — THRUST OF SPIRAL TEETH. 



The oblique action of the teeth of 
spiral gears on each other tends to throw 
the gears bodily in the direction of their 
axes, and this tendency creates a thrust 
that must be opposed by thrust bear- 
ings. The end pressure on the shaft of 
a worm is greater than that exerted on 
the teeth of the worm gear it is driving. 

When the shafts are parallel the 
thrust may be completely avoided by 
the use of double spiral or "herring- 
bone " teeth, Fig. 82 or 83, which act in 
opposite directions, and neutralize each 
other. 

When the shafts are at right angles 
the thrust may be neutralized by op- 
posing a second gear in the manner 
shown in section by Fig. 95. The two 
worms with opposite spirals run in two spiral 
worm gears that also work with each other, 
and, as the pressure on one gear is opposite 
that on the other, there is no thrust on the 
shaft. When this combination is made with 
worm gsars having concave teeth, the teeth 
can bear only at their ends. 




Arrangement to avoid tJirust 

Fig. 95. 



When the thrust cannot be avoided it 
should be taken by a roller bearing, rather 
than by the common collar bearing. The 
diagram, Fig. 94, shows the superior efficiency 
of the roller bearing as compared with the 
collar bearing, the gain being from ten to 
twenty per cent. 



114. — THE HOBBED OR CONCAVE WORM GEAR. 



If a spiral gear is made of steel, provided 
with cutting edges by making slots across its 
teeth, and hardened, it will be a practical 
cutting tool called a spiral milling cutter or 
hob. Fig. 96 shows a spiral milling cutter, 
having a great spiral angle, and therefore 
called a worm. 

If this cutting spiral gear is mounted in 
connection with an uncut blank so that both 
are rotated with the proper speeds, and the 
shafts of the two gears are gradually brought 
together while they are revolving, the edge 
of the blank will be formed with concave 
teeth that curve upwards about the sides of 
the cutting gear. If the hob is then replaced 
with a spiral gear that is a duplicate of it, ex- 
cept that it has no cutting teeth, we shall 
have the familiar worm and worm gear of 
Fig. 97. 

The principle of the concave gear applies 
to any pair of spiral gears, on shafts at any 




Concave Worm Gecvr and Worm. 
Fig. 97. 



CO 



Worm Gearing. 



angle, but in practice it is confined to the 
worm and gear on shafts at right angles. 

The nature of the contact between the 
worm and the concaved worm gear has not 
yet been definitely determined, but there is 
no reason to suppose that it is different from 
that between plain spiral teeth, a point con- 
tact on the normal spiral, but it is probably 
continuous. It is certain, however, that the 
contact is considerably closer, more nearly 
resembling surface contact, and being sur- 
face contact when the diameter of the gear is 
infinite. 

The worm is adjustable in the concaved 
teeth of the gear in the direction of its axis, 




A Hob. Fig. 96. 

aid will change its position as required by 
die wear of the thrust bearing. It is not ad- 
justable laterally. 



115. — HOBBING THE WORM GEAR. 



When the hob is provided it is a simple 
matter to cut the gear. The gear is generally 
provided with the desired number of notches 
in its edge, that are deep enough to receive 
the points of the teeth of the hob, and 
the hob will then pull it around as it 
revolves. 

It is a too common practice to make the 
hob do its own nicking, for, if it is forced 
into the face of the gear as it revolves, it will 
pull it around by catching its last teeth in 
the nicks made by the first. 

If luck is good these nicks will run into 
each other, and the gear will be cut with 
teeth that appear to be correct, but, as the 
outside diameter is greater than the pitch 



diameter, there will be one, two, or three 
teeth too many. The teeth of the finished 
gear are therefore smaller than those of the 
worm by an amount that is ruinous if the gear 
is small, although it is not noticeable when 
the diameter is large. If there are 12 teeth 
where there should be but 10, each tooth will 
be too small by two-twelfths of itself; but if 
there are 102 teeth where there should be but 
100, ea> h 100th is too small by but two- 
hundredths of itself. This handy makeshift 
proce e s will do very well on large ge^rs, but 
not on small ones, unless the worm to run in 
the gear is made to fit the tooth, with a tooth 
that is smaller, and lead that is shorter than 
that of the hob. 



116.— INVOLUTE WORM TEETH. 

Worms are generally cut in the lathe, and 



as a straight-sided tooth is most easily 
formed, the involute tooth is generally 
adopted. 

Strictly, the form of the tool should be 
that of the normal section of the thread, and 
it should always be set in the lathe with its 
cutting face at right angles to the thread. 



But custom and convenience allow the tooth 
to have s raight side?., and to be set with its 
face parallel with the axis of the worm, and 
the real difference is not generally notice- 
able. 

The standard tool has its sides inclined at 
an angl 1 of 30°, and has a length and a width 
dependent upon the pitch. 



117. — INTERFERENCE OF INVOLUTE WORM TEETH. 



There is one difficulty that is seldom recog- 
nized, but which must be carefully guarded 
against if properly running gears are ex- 



pected, and that is interference. The teeth 
of worm gears will interfere with each other 
when the conditions are right for interference. 



Worm Gearing. 



61 



just as spur involute teeth will interfere, as 
shown by Ficr. 36. Fig. 98 shows the gear 
that would be formed by the usual process. 
The difficulty can be remedied by rounding 



over the tops of the teeth of the hob and 
worm, as described in (55). 

It is also a simple matter to avoid the inter- 
ference by enlarging the outside diameter of 




Interfering Worm. 
Fig. 98. 



Interference Avoided. 
Fig. 99. 




pt---Y — n~ 



Involute worm and gear 
twenty-one or more teeth 

Fig. 100. 



62 



Worm Gearing. 



the worm gear. If, as shown by Fig. 99, the 
tooth has but a short flank, or none at all, 
and the addendum of the gear is about twice 
that by the usual rule, the action will be con- 
fined to the face of the gear and the flank of 
the worm, and there can be no interference. 
By adopting an obliquity greater than 15°, 
interference can be avoided without changing 
the addendum. 

This method has the advantage that the 
same straight-sided worm and hob can be 
used for small gears as for large ones, and 
the disadvantage that the action is confined 
to the approach and subject to greater fric- 
tion (48). 

When the standard 30° tool is used, all 
gears of 26 teeth, or smaller, should be made 
in this way, but the correction is not strictly 
necessary for gears of more than 20 teeth, 
unless particularly nice work is required. 

Fig. 100 shows the proper construction of a 
gear of 21 or more teeth, and Fig. 101 shows 
that of a gear of less than 21 teeth. In the 
former case, the teeth of the worm should be 
limited by the limit line 11, but the interfer- 
ence for 21 to 25 teeth is not noticeable. 



Draw worm teeth straight 
Draw gear teeth by points (57) 





Involute worm 
for twenty or 



and gear 
less teeth 

Fig. 101. 



118. — CLEARANCE OF WORM TEETH. 



There is another practical point that is sel- 
dom recognized, and that is that worm teeth 
should have clearance (40), for there is no 
reason for clearing spur teeth that will not 
apply quite as well to any other kind. 

The clearance is easily obtained by making 
the tooth of the hob a little longer than that 
of the worm, as shown by the tooth a of Fig. 



100. For the same reason the hob should 
have no clearance at the bottom of its thread, 
so that the tops of the gear teeth will be 
formed of the proper length. The custom 
of making the hob and worm of exactly the 
same diameters will apply only when the 
worm "bottoms" in the gear and the gear 
bottoms in the worm. 



119. — CIRCULAR PITCH WORM TEETH. 



The old and clumsy circul ar pitch system 
is in universal use for worm teeth, for the 
reason that worms are generally made in the 
lathe, and lathes are never provided with the 
proper change gears for cutting diametral 
pitches. The error is so firmly rooted that it 
is useless to attempt to dislodge it. 

It is therefore necessary to figure the diam- 
eters of worm gears as if their throat sections 



were the same as those of common spur gears 
and racks on the circular pitch system. The 
table of diameters (35) will be of great assist- 
ance. 

One great objection to the use of the circu- 
lar pitch system for spur gears does not ap- 
ply to worm gears, that the center distance 
between the shafts will always be an incon- 
venient fraction, unless the pitch is as incon- 



Diametral Worm Gearing. 



63 



venient. The worm can be made of any 
diameter, and can therefore be made to suit 
the pitch diameter of the gear and the center 
distance at the same time. 

The sides of the tool for circular pitches 
should come together at an angle of thirty 
degrees, and the width of the point, as well 
as the depth to be cut in the worm or in the 
hob, should be taken from the following 
table. The diameter of the hob should be 
greater than that of the worm by the "in- 
crease" given. 

Make the tool with the proper width at the 
point to thread the worm, and then, after 
making the worm, grind off half the " in- 
crease" from the length of the tool, and use 
it to thread the hob. 



TABLE FOR CIRCULAR PITCH WORM TOOLS. 



Circular pitch 

Point of hob tool 

Point of worm tool. . . 
Depth of cut in worm 

or hob 

Increase 

Circular pitch 

Point of hob tool 

Point of worm tool . . . 
Depth of cut in worm 

or hob 

Increase 

Circular pitch. 

Point of hob tool 

Point of worm tool . . . 
Depth of cut in worm 

or hob 

Increase 



.644 
.620 

1.416 
.165 



.322 
.310 

.708 
.0b3 



.141 
.135 

.310 
.036 



m 

.564 
.542 

1.240 
.146 



.282 
.271 

.620 
.073 



.121 
.116 

.265 
.031 



1V6 

.483 
.466 

1.062 
1.249 



.241 
.233 

.531 
.062 



100 
.097 

.222 

.026 



.402 .362 
.388 .349 

.886 .797 
.104.. 094 



.201 
.194 

.443 
.052 



M 

.080 
.078 

.177 
.021 



.161 
.155 

.354 
.042 



A 

.060 
.058 

.133 
.016 



120. — DIAMETRAL PITCH WORM TEETH. 



If the proper change gears are provided, it 
is as easy to cut diametral pitch worm teeth 
as any. The proper gears can always be 
easily calculated by 'the rule that the screw 
gear is to the stud gear as twenty-two times the 
pitch of the lead screw of the lathe is to seven 
times the diametral pitch of the worm to be cut. 

For example, it is required to cut a worm 
of twelve diametral pitch, on a lathe having 
a leading screw cut six to the inch. We have 
screw gear _ 22 X 6 _ 11 
stud gear 7 X 12 ~ ~7~ ' 

and any change gears in the proportion of 11 
and 7 will answer the purpose with an error 
1 



of 



of an inch to the thread of the worm. 



10,000 

If 22 and 7 give inconvenient numbers of 
teeth, the numbers 69 and 22 can be used 
with sufficient accuracy, and 47 and 15, or 
even 25 and 8 may do in some cases. 

To save calculation and study, the table of 
change gears for diametral pitches is provided, 
and it will give the proportion of screw gear 
to siud gear to be used for all ordinary cases. 

The pair on the left will give the proper 
pitch within less than a thousandth of an 
inch, and that on the right will serve with an 
error always less than a hundredth of an inch, 
and sometimes less than two or three thou- 
sandths of an inch. 

Having the change gears, figure the pitch 



diameter of the gear as if the throat section 
is a spur gear on the diametral pitch system. 
The sides of the tool should come together 
at an angle of thirty degrees, and the width 
of the point of the tool, as well as the depth 
to be cut in the worm or in the hob, should be 
taken from the following table. The diame- 
ter of the hob should be greater than that of 
the worm by the ' ' increase " given. 

TABLE FOR DIAMETRAL PITCH WORM TOOLS. 



Diametral pitch. 



Point of hob tool 

Point of worm tool 

Depth of cut in worm or 

hob 

Increase 



Diametral pitch. 



Point of hob tool 

Point of worm tool.. . 
Depth of cut in worm or 

hob 

Increase 



Diametral pitch. 



Point of hob tool 

Point of worm tool 

Depth of cut in worm or 

hub 

Increase 



1 


2 


3 


1.035 
.968 


.517 

.484 


.345 
.323 


2.125 
.250 


1.063 
.125 


.708 
.083 


5 


6 


7 


.207 
.194 


.173 
.162 


.148 
.138 


.425 
.050 


.354 
.042 


.304 
.036 

14 


10 


12 


.104 
.097 


.086 
.081 


.074 
.069 


.213 

.025 


.177 
.021 


.152 
.018 



.258 
.242 

.532 
.063 



.129 
12-1 

.266 
.032 

16 

.065 
.060 

.133 
.016 



Make the tool with the proper width at 
the point to thread the worm; and then, 
after making the worm, grind off half the 
"increase " from the length of the tool, and 
use it to thread the hob. 



64 



Dlajnetral Worm Gea-ring. 



Pitch of Leading Screw. 



o 

© 
.a 





o 



CO 

© 

£ 

5 



3 
4 
5 
6 

8 

10 
12 
14 
16 



2 3 



6 



8 10 



44 23 
21 11 


22 

7 


88 46 
21 11 
22 

7 ' 


110 
21 


44 69 
7 11 
33 

7 ' 


21 
3 ' 


176 92 
21 11 

44 

7 ' 


220 
21 ' 


11 

7 ' 


33 
14* 


55 
14* 
22 25 
7 8 
35 50 
21 19 


11 
2 ' 


55 

7 ' 

44 25 
7 4 


44 5 
35* 4 
22 
21* 


66 
35 

11 

7 


15 

8 
30 
19 


88 5 
35 2 


132 15 
35 4 
22 60 
7 19 


22 35 
5 8 


176 5 
35 1 
88 80 

21 19 
176 18 
49 ' 5 

22 16 
7 5 


44 40 
21 19 
88 9 
49* 5 


11 70 

3 19 
22 63 
7 *20 
11 14 

4 5 


110 100 
21 ' 19 


44 9 
49* 10 


66 
49' 
33 
28 


27 
20 
6 
5 


110 9 
49 ' 4 


132 27 
49 *10 
33 12 
14 5 


220 9 
49 ' 2 


11 4 
14 5 
22 5 
35 8 


11 8 
7 5 


55 2 
28 1 


55 4 
14 1 


33 
35 


15 
16 


44 5 
35 4 


11 25 
7 16 
55 25 
42 19 


66 15 
35 8 


11 35 
5 16 


88 5 
35 4 


22 25 

7 8 


11 10 
21 19 


11 
14 


15 
19 


22 20 

21 19 

44 8 
49 9 

22 4 

28 5 


11 30 
7 19 
66 4 
49 3 


11 35 
6 19 


44 40 
21 19 


55 50 
21 19 


22 4 
49 9 


33 
49 


2 
3 


55 10 
49 9 

55 1 

56 1 


11 14 

7 9 


88 16 
49 9 


110 20 
49 9 


11 2 

28 5 


33 

56 


3 

5 


33 6 

28 5 


77 7 
56 5 


11 8 

7 5 


55 2 

28 1 



Exact numbers on the left. Approximate on the right. 
Table op Change Gears for Diametral Pitch Worms. 



121. — WIDTH OP WORM GEAR FACE. 



The bearing between the tooth of the 
worm and that of the gear is near the center 
of the gear, and it is quite small (104). It is, 
therefore, useless to make the gear with a 
wide face. If the face is half the diameter 
of the worm it will have all the bearing that 
can be obtained, and any extra width will 
simply add to the weight and cost of the gear. 



The length of the worm need be no more 
than three times the circular pitch, for there 
are seldom more than two teeth in contact at 
once. If, however, the worm is made long, 
it can be shifted when it becomes worn, so 
as to bring fresh teeth into working position. 
This provision is wise, for the reason that the 
worm is always worn more tban the gear. 



122.— THE HINDLET WORM AND GEAH. 



If the cutting hob and the worm is shaped 
by the tool a, and the process indicated by 
Fig. 102, the resulting pair of gears is known 
as the Hindley worm and gear. The worm 
is often called the " hour glass " worm. 



It is commonly but erroneously stated 
that this worm fits and fills its gear on the 
axial section, the section that is made by a 
plane through the axis of the worm and 
normal to the axis of the gear. It has even 



Hindley Worm Gearing. 



65 



"been stated that the contact is between sur- 
faces, the worm filling the whole gear tooth. 

The real contact is not yet certain, but it is 
certain that it is not a surface contact. It is 
also certain that it is on the normal and not 
on the axial section, and that the Hindley 
worm hob will not cut a tooth that will fill 
any section of it. The contact may be linear, 
along some line of no great length, but it is 
probably a point contact on the normal sec- 
tion. The order of the contact is certainly 
very close, resembling that of two surfaces. 

The worm is limited in length, for the 
sides of the teeth cannot slant inward from 
the normal to the axis. The end tooth m in 
Fig. 102 cannot be used, for it will destroy 
the teeth of the gear as it is fed towards 
this axis in the operation of hobbing. 

It has the one great defect that it is not 
adjustable in any direction, and, therefore, 
cannot change its position when the shaft 



T1i€ Hindley 
Worm Gear 




-Fi{/. 102 



bearings wear, unless it is itself worn the 
same amount. It is doubtful if this form of 
gearing has any advantage over the plain 
spiral gearing, except when new and in per- 
fect adjustment. 



123. — THE PEST WORM AND GEAR. 



If the hob and the worm are shaped by 
the pin-shaped revolving milling tool b of 



JPin worm gear. 

Fig. 103. 




Fig. 102, the gearing produced will have 
linear bearing between the teeth. 

The action will be the same as between a 
series of pin teeth like the milling tool, each 
pin being in the axial section of the worm, 
but having a linear bearing on the normal 
section of its teeth. 

This form of gearing, which is a 
modification of the Hindley form, 
may take the shape of pin gearing, 
the* teeth being round pins like the 
milling tool. If the pins are mounted 
on studs, so as to revolve, a roller pin worm 
gear will be produced. 

Fig. 103 shows a form of roller pin 
gearing in which the pins have been en- 
larged. 



124. — THE WHTTWORTH HOBBTNG MACHINE. 



When the amount of work to be done will 
warrant the use of a special machine, the 
hobbing machine of Sir Joseph Whitworth 
mgy be u-ed. It was invented in 1835, and 
has not been materially improved since then, 



although there are numerous patents relating 
to it. The worm gear to be hobbed is fixed 
upon the same spindle with a master worm- 
wheel. A driving worm runs in the master 
wheel, and it is connected by a train of gear- 



66 



Hobbing Machines. 



ing with a hob that is so mounted on a 
carriage that it can be fed towards the gear 
blank. The hob is slowly forced into the 
blank, while both are revolving with the 
proper speeds, and the gear is cut without 
the assistance of previously made nicks. See 
British patent 6,850 of 1835. 




Fig. 106 
Spiral and Spur Gear. 



125. — THE CONJUGATOK. 



This is a machine for cutting spur or spiral 
gears by means of a hob, and its principle is 
an extension of that of the Whitworth worm 
gear hobbing machine. 

If, when the hob in the Whitworth ma- 

Conjugator. Elevation 

Fig. 104. 




Plan 

Fig. 105. 



chine has reached the full depth of the tooth, 
it receives a new motion in the direction 
of the tangent to its pitch spiral, it will 
continue the tooth to the edge of the gear, 
and form the plain spiral gear of Fig. 91. 

Fig. 104 is an elevation of the machine, and 
Fig. 105 is a plan. The hob h is mounted 
upon an arbor that is connected by a train of 
gearing with the spindle s that carries the 
blank gear g to be cut, so that the hob and 
blank revolve together with any definite 
proportionate speed. The hob is carried 
upon a carriage that is fed on a frame /. 
The hob swivels upon the carriage, so that 
the tangent to its pitch spiral can be set 
parallel with the direction of the feed, and 
the frame swivels so that the tooth can be 
cut at any angle with the gear spindle. 

As the blaDk and the hob are revolving, 
the latter is fed into the former, and it will 
cut a perfect tooth in the direction that the 
frame is set at. As the frame can be set in 
any direction, the machine will cut the com- 
mon straight tooth, as shown by Fig. 106. 
All gears cut by the same cutter will run 
together interchangeably, and if two spiral 
gears are cut at the same angle in opposite 
directions they will run together on parallel 
shafts. See U. S. patent number 405,030 
June 11th, 1889. 



7. IRREGULAR AND ELLIPTIC GEARS. 



126. — NON-CIRCULAR PITCH LINES. 



The consideration of pitch lines that are 
not circular, and of the teeth that are fitted for 
them, is an interesting but not particularly 
important branch of odontics. Such pitch 



lines are largely used for producing variations 
of speed and power, but have no other prac- 
tical applications. 



127. — THE IRREGULAR PITCH LINE. 

The most general case is that of two indefi- 
nite irregular curves rolling together, Fig. 
107, the only condition being that they 
shall be so shaped that they will roll together 
continuously. 

As the practical importance of the free 
pitch line is very small, we shall not ex- 
amine it in detail. 




Irregular pitch, lines 

Fig. 107* 



128. — PITCH LINES ON FIXED CENTERS. 



When we attach the condition that the two 
pitch lines shall revolve in rolling contact on 
fixed centers, we have a definite problem of 
more interest and importance than that of the 
free pitch line. 

If, as in Fig. 108, we have a pitch line A 
revolving upon a fixed center a, we can con- 
struct a pitch line B that will roll with it, 
and revolve on the given fixed center b, by 
the following process. 

From any pitch point 0, step off equal arcs 
Oc, cc, cc ; draw circular arcs cd from the 
center a; draw circular arcs dn from the 
center b; step off the same equal arcs Oe, 
ee, ee, then Oeee will be the required mat- 
ing pitch line. 

These curves will always be in rolling con- 
tact at a point on the line of centers ab, the 
pitch point and the angle of the curves with 
the line of centers continually changing. 

The velocity ratio of the curves will be 



nn 




Fixed centers 

Fig. 108. 

variable, and always equal to the inverse pro- 
portion of any two mating radiants, ac and be. 



68 



Multilobes. 



129. — CLOSED PITCH LINES. 

When one of the curves of Fig. 108 is a 
closed curve, the other will in general not 
be closed, but by trying different centers, a 
curve can be found that will be closed. 

If the closed curve a,, Fig. 109, is taken, 
the mating curve A x will be closed when the 
center is chosen at a certain point B lt that can 
be found by repeated trials. 

The mating closed curves thus constructed 
will seldom be alike, but will always have 
points of similarity. A salient point q on one 
will be paired with a reversed point or notch 
on the other, and lobes on one will be repre- 
sented by depressions on the other. Half a 
revolution of one of the curves, from any 
position, will turn the other through half a 
revolution. 




Set of Multilobes 

Fig. 109. 



130. — MULTILOBES. 



If, after finding the center B 1} Fig. 109, for 
the closed mating curve, other centers are 
tried, second, third, and succeeding centers, 
B 2 , B s , B±, will be found, about which the 
mating curves will also be closed. 

These closed curves, called multilobes, will 
be each divided into like lobes, the second 
curve, or bilobe, into two lobes ; the third, or 
trilobe, into tbree lobes, and so on. 

If the center is placed at infinity, the rack 
lobe A qo will be formed. 

If the center be taken negatively, on the 
same .side as the original center b lf at b 2 , 
b 3 , b A , etc., negative multilobes a.>, a 3 , a A , 
etc., will be formed about the original curve 
a t . 

All these multilobes, positive and negative, 
will roll together collectively about their fixed 
centers, in rolling contact at a common and 
shifting pitch point O. 

Any two, of the same sign, will roll in inter- 
nal contact, and any two of opposite signs 




Train of multilobes 

Fig. 110. 

will roll in external contact, so that they can 
be formed in train, Fig. 110. 

When it so happens, as it does with the 
ellipse revolving on its focus, or the logarith- 
mic spiral revolving on its pole, is taken, that 
the first derived pair of curves, or unilobes, 
are exactly alike, all the multilobes will be 
alike ; the positive trilobe like the negative 
trilobe, and so on, so that any two curves of 
such a set will work together in either inter- 
nal or external contact, Fig. 111. 



Conic Pitch Lines. 



69 



131. — CONIC SECTION PITCH LINES. 



If two like conic sections are mounted upon 
their foci, they will roll together. 

Their free foci will revolve at a fixed dis- 
tance from each other, and may be connected 
by a link. The line of the free foci will in- 
tersect the line of the fixed foci at the point 
of contact of the pitch lines. 

Fig. 112 shows a pair of ellipses, Fig. 113 
a pair of parabolas, and Fig. 114 a pair of 
hyperbolas. 

The elliptic pitch line is the only one 
known that will revolve with its equal, and 
make a practical and complete revolution. 



fixed 





Elliptic multllobes 

Fig. 111. 



Elliptic pitch lines 

Fig. 112. 




Parabolic pitch tinea 
Fig. 113. 




'ocus 



Hyperbolic pitch lines 



70 



Logarithmic Pitch Lines. 



132. — THE LOGARITHMIC SPIRAL. 



n = 



If the radiants a, b, c, d, e, Fig. 115, make 
equal angles with each other, and each one 
is equal to the adjacent one multiplied by a 
constant number, their extremities will deter- 
mine a logarithmic spiral. 

If the first radiant a is given, with the con- 
stant multiplier n, the second radiant will be 
na, the third will be n % a, the fourth will be 
n s a, and so od. 

If the first and last radiants, a aDd e, are 
given, and there are p equal angles between 
them, the constant is 

_ P/~e 

-v ? 

so that it is a simple matter to construct a 
logarithmic spiral to connect any two given 
radiants at any given angle with each other. 

The curve possesses the singular property 
that all tangents, A or E, make the same 
angle with the radiants at their points of con- 
tact. The curves are always inclined to the 
line of centers at the constant angle. 

The curve continually approaches the 
center M, or "pole," making an infinite 
number of turns about it, but never reach- 
ing it. 

It also has the eniirely useless property that 
the pole will trace an involute of the base 
circle if it is rolled upon the pitch circle (75). 

It possesses the property, not possessed by 
any other curve, that it will roll with an 
equal mate on fixed centers that can be varied 
in position. The curve H will roll with the 
curve C, whether its pole is at jV, or at 8, or 
at 7. 

Fig. 116 shows a pair of logarithmic spi- 
rals in internal contact. 




Ziogarithmic pitch lines 

Fig. 115. 




Tntemail logarithmic 
pitch'lines 

Fig. 116. 



133. — COMPOSITE PITCH LINES. 



Instead of drawing a curve at random, 
and finding the mate to run with it, Fig. 
108, the complete pitch line may be built 
up of a number of curves, of which the 
properties are known. 

Thus, Fig. 117 shows composite gears, 
consisting of circular parts J. and a, and an 
elliptic trilobe B, working with an elliptic 



bilobe b. Fig. 118 shows a combination of a 
pair of logarithmic spiral arcs A and a, a 
pair of elliptic bilobal arcs B and b, a pair 
of logarithmic spiral arcs D and d, and a 
pair of elliptic quadrilobal arcs E and e. 
An endless variety of combinations can be 
made in this way. 
It is not necessary that the component 



Composite Pitch Lines. 



71 



curves be tangent, if they succeed each other 
continuously. Fig. 119 shows a pair of equal 
logarithmic spirals with a break at ab, the 
action at b commencing just as it ends at a. 
Care should be taken to avoid salient 




Composite pitch lines 

Fig. 117. 



points, breaks, and interruptions of the con" 
tinuity of the curve, for there must be 
defective tooth action at such points. The 
curves should run smoothly into each other 
with gradual changes of curvature. 




Composite pitch lines 

Fig. 118. 




Broken pitch lines 

Fig. 119. 



134. — TEETH OF NON-CIRCULAR PITCH LINES. 



The action of the teeth of non-circular 
pitch lines does not, at first sight, appear to 
follow the laws pertaining to circular lines, 
but there is really very little difference. 

If we consider the two pitch lines to be 
free, and to be so moved while they roll 
together that the pitch point 0, Fig. 107, is 
fixed, and so that the fixed line c G is always 
at right angles to both curves at their com- 
mon point 0, the laws of the tooth action will 
be almost precisely the same as laid down 
for the circular pitch line. Fig. 107 may be 
easily applied to (24) as illustrated by Fig. 15. 

When the centers are fixed, the same tooth 



action takes place, but the line of action and 
the pitch point continually change their 
positions. 

The teeth of non-circular pitch lines can 
therefore be formed either by conjugating a 
given odontoid, as in (24), or by the rolled 
curve theory of (32). 

By all means the most practicable method, 
when the circumstances will permit, is to 
make up the curve by joining approximating 
circular arcs, and to provide each circular 
arc with teeth in the ordinary way. See this 
process as applied to the elliptic pitch line at 
Figs. 129 and 130. 



135. — TEETH AT SALIENT POINTS AND BREAKS. 



When there is a salient point, or other inter- 
ruption of the continuity of the action, as at q, 
Fig. 109, or at Mm, Fig. 118, there must be 



an interruption in the arrangement of the 
normals of any tooth curve, and a consequent 
failure of the tooth action. 



12 



Elliptic Pitch Lines. 



Fig. 120 shows a cycloidal tooth curve M, 
at a corner or salient point 8, between two 
circular pitch arcs. There is a circular arc 
A on the odontoid made while the describing 
circle is turning about the point S, and that 
arc can have no continuous tooth action. 
Therefore the tooth action will fail, unless 
the next tooth curve iV springs from the 
salient point. 

If a tooth springs from the salient point, 
the tooth action will be correct, but mechani- 
cally imperfect, as the arc of action of two 
teeth cannot lap over each other to allow for 
practical defects. And then, as two tooth 
curves cannot spring from the same pitch 
point in opposite directions, such gears can 
run in but one direction, and are not reversi- 
ble. 

When there is a break, as at db, Fig. 119, 
the teeth must be so cut off that they will 




The salient point 

Fig. 120. 



separate at a just as they engage at b, for there 
is a sudden change in the velocity ratio. 
Such combinations are practicable, but in 
every way undesirable. 



136. — THE ELLIPTIC GEAK. 



The principal, and almost the only use of 
the irregular gear, is to produce a variation 
of speed between certain given limits, with- 
out conditions as to the variations of speed 
and details of the motion between the limits. 
"When that is the only object, the elliptic 
pitch line is the only one that is required, and 
it is chosen because it is the only known con- 
tinuous closed curve that will work in roll- 
ing'contact with an equal mate, and because 
it is, next to the circle, the simplest known 
curve. Of the elliptic multilobes, the uni- 
lobe, or simple ellipse, revolving on one of 
its foci as a center, is the only one used to 
any appreciable extent, and therefore is the 
only one that requires examination in detail. 

The use of the elliptic gear is practically 
confined to producing a simple variation of 
speed between known limits, and to produc- 
ing a ' ' quick return motion " for planers, 
shapers, slotters, and similar cutting tools, as 
well as for pumps, shears, punches, shingle 
machines, and others where the work is done 
mostly during one-half of the stroke of a 
reciprocating piece. The work of a planer 



tool or of the plunger of a single acting 
pump, is all done during the motion of the 
tool or of the plunger in one direction, and 
the only object on the return is to get the 
piece ready for the next useful operation in 
the quickest possible time. 

For an example, the bobbin of a spinning 
machine is to be wound in a conical form, 
the thread being fed to it through a moving 
guide, and the necessary variable motion of 
the guide, fast at the point of the cone, and 
slow at its base, is best given to it by a pair 
of elliptic gears. For another example, the 
motion of the platen of a printing press 
should be rapid when the press is open, and 
slow and powerful when the impression is 
being taken, and the object can be reached 
best by a pair of elliptic gears operating the 
platen. 

The practical uses of the elliptic gear are 
endless, and it would be in greater use and 
favor, if it were not for the fact that its pro- 
duction, by the means ordinarily in use for 
that purpose, is as difficult and costly as the 
resulting gear is unsatisfactory. 



Elliptic Pitch Lines* 



137.— THE 

To thoroughly understand the construc- 
tion and operation of the ellipse, it is neces- 
sary to learn but a few of its many proper- 
ties. 

The mechanical definition of the ellipse is 
that it is one of the " conic sections." If the 
cone, Fig. 121, is cut by a plane G at right 
angles with its axis, the outline of the section 
will be a circle; if the plane incuts the cone at 
an angle, the section will be an ellipse; if the 
plane P is parallel with the side of the cone, 
the section is a parabola, and if the plane H 
is at such an angle that it cuts both nappes of 
the cone, the section is a hyperbola. All 
these curves will roll together when fixed on 
centers at certain points called foci, but the 
ellipse, and its special case, the circle, are the 
only ones that are capable of continuous mo- 
tion. 

In the ellipse, Fig. 122, the point G is the 
center, the longest diameter, A A' ', is the 
major axis, the shortest diameter, BB , is the 
minor axis; A and A' are the major apices, 
and B and B' are the minor apices. 

If an arc be drawn from the minor apex, 
with a radius equal to the major semi-axis, it 
will cut the major axis at points F and F, 
called the foci, and one focus must be chosen 
as the center, about which the curve is to re- 
volve if used as the pitch line of a gear. 

It is a property of the curve that the sum 
of the distances, PF and PF' , frcm any 
point to the foci is equal to the major axis, 
AA', and this feature is used as a means of 
constructing the curve by points. Draw any 
arc at random from one focus with radius 
FP. Draw an arc from the other focus with 
a radius equal to A A' — FP, and it will cut 
the first arc at a point of the ellipse. When 
the point P is near either major apex, the 
arcs intersect at such a sharp angle that the 
method is nearly useless. 

Another, and much the best known method 
for constructing the ellipse by points, is to draw 
any radial line L, and also circular arcs W 
and V, from the center throngh the apices. 
From the intersections, w and v, of the radial 
line and the circles, draw lines parallel to the 
axes, and they will intersect, always at right 
angles, at a point u on the curve. This 



ELLIPSE. 




Conic sections 

Fig. 121. 




The ellipse 

Fig. 122. 



method is very accurate, and has no failing 
position. 

Another valuable property of the ellipse is 
that if the line pab be so drawn that the dis- 
tance pa is equal to BG, and pb to A G, the 
point p will be upon the curve if the points a 
and b are upon the axes. 

The curvature of the ellipse is an important 
feature in connection with its use as a gear 
pitch line. It is sharpest at the major axis 
A, and flattest at the minor apex B, else- 
where varying between the two limits. 

The radius of curvature at either apex, 
that is, the radius of the circle that most 
nearly coincides with the curve, is found by 
drawing the lines Bh and Ak at right angles 



Elliptic Pitch Lines. 



with the chord AB. The distance Gh is the 
radius of curvature at the major apex A, and 
the distance Gh is the radius at the minor 
apex B. 



The normal PN to the curve at any point 
P bisects the angle FPF' between the focal 
lines, and the tangent PT is at right angles 
to the normal. 



138. — ELLIPTOGRAPHS. 



There are a multitude of elliptographs, or 
instruments for drawing the ellipse, but only- 
two of them are of practical application in 
this connection. 

The simplest known elliptograph consists 
of a couple of pins, a thread, a pencil, and a 
stock of patience. The pins are inserted at 
the foci, as in Fig. 123, and the curve is 
drawn by moving the pencil with a uniform 
strain against the string. After a number of 
trials, depending in number on the skill of the 
draftsman, the curve may be induced to pass 
through the desired points. The best result 
will be obtained by the use of a well waxed 
thread running in a groove near the point 
of a hard pencil. The pencil should be long, 
and held by the end, so that the strain on the 
string will be uniform, for the elasticity of 
the string is the greatest source of 
error. This "gardener's ellipse" 
will generally be accurate enough 
for a tulip patch, but should not 
be relied upon for mechanical pur- 
poses, unless one or more points 
between the apices are tested and 
found to be correct. If the two 
pins and the pencil are circular, 
and of the same diameter, the ac- 
curacy of the ellipse is independent 
of their diameter. 

The best elliptograph is the 
" trammel," Fig. 124, which takes 
a variety of shapes, but which in 
its simplest condition consists of 
a cross, with two grooves at right 
angles, and a bar D with two pins 
a and b, and a tracing point P 
placed in line. The distance Pb 
being set to the major semi-axis, and 
the distance Pa to the minor semi- 
axis, the point P will trace the ellipse if the 
pins are confined to move in the grooves. 
If carefully made, the instrument works 



with great precision, is easily handled and 
set, and, if the curve drawn is not very 
flat, it may be inked. The cheap wooden 



Fig. 123. 




Gardener's ellipse. 




The trammel 

Fig. 124=. 



trammel should not be tolerated, for the 
string and two pins cost less and are more 
reliable. 



Elliptic Pitch Lines, 



t>) 



139. — APPROXIMATE CIRCULAR ARCS. 



If a well-made trammel is not at hand, the 
best plan is to draw the ellipse with a string, 
through several constructed points, and then 
to ink it by finding centers for approximate 
arcs, as shown by Fig. 125. An arc from a 
center m on the major axis, will coincide 
very well with the curve near the major 
apex, a similar arc n from a center on the 
minor axis will serve near the minor apex, 
and a third center q can be found for an arc 
to join the first two. More than three cen- 
ters will seldom be -required, and when the 
ellipse is not very flat the two centers on the 
axes will be sufficient. 




The elliptic involute 

1 



140. — FOUR CENTER ELLIPSE. 



When the ratio of the axes is not less than 
eight to ten, as is generally the case, a prac- 
tically perfect ellipse 'may be drawn from 
four centers by the following method. 

Draw the line GL, Fig. 126, parallel to 
A'B, and construct the point u on the ellipse 
by the method of (137). Find a point a on 
the major axis, from which an arc from A 
will pass through u, and it will be the major 
center. It may be found by trial, or by 
drawing um at right angles to uA, and 
bisecting Am in a. 

Through a draw ac at right angles to AB, 
and its intersection with the minor axis will 
be the minor center b. Lay off Ca' and Gb' 
equal to Ca and Gb, and draw be' , b'c", and 
b'c"'. 

From the centers a draw the arcs cAc'" , 
and c'A'c", and from the centers b draw the 
arcs cBc' and c"B'c'". 



Fig, 126. B 




Tour center method 



Lines that are parallel to the pitch line, 
such as the addendum, root, clearance, and 
base lines, are to be drawn from the same 
centers. 



141. — ROLLING ELLIPSES. 



When two equal ellipses, Fig. 127, are 
arranged to revolve on their foci as centers, 
with a center distance equal to the major 
axis, they will roll together perfectly, and 
be fitted to act as the pitch lines of gear 
wheels. 

When the driver D turns in the direction 



of the arrow d, it will drive the follower 
i^by direct contact of the pitch ellipses, but 
when turning in the other direction with 
respect to the follower, as it must during 
half of its revolution, it has no direct 
driviDg action, and the follower must be 
kept in contact by some other force. 



76 



Spacing the Ellipse. 



As the two ellipses roll together, the free 
foci F 2 and F 4 will always move at a con- 
stant distance apart, equal to the distance 
between the fixed foci, and therefore they 
may be connected by the link L. 

The center line of the link will always 
cross the fixed center line at the point of con- 
tact of the ellipses, and the tangent T at that 
point will pass through the intersection of the 
axes. 




Rolling ellipses. 

Fig. 127* 



142.— SPACING THE ELLIPSE. 



As the ellipses roll together it is essential 
that the axes come in line, and therefore, if 
the teeth of one gear are fixed at random, 
those of the other must be fixed to corre- 
spond. If this requirement is satisfied, it 
makes no difference where the teeth are 
placed. 

It is, however, very desirable that the two 
gears shall be exactly alike, so that they 
can be cut at one operation while mounted 
together on an arbor through their focus 
holes, and to do this, it is necessary to start 
the teeth at different points, according to 
whether their number is odd or even. 

If the number of teeth is even, one tooth 
must spring from the major axis, as shown 
by Fig. 128. 

If the number of teeth is odd, the major 
axis must bisect a tooth and a space, as shown 
by Fig. 129. In this case, if one of the 
gears can be turned over, or if its other focus 
hole can be used as a center, it may have a 
tooth springing from the major axis. 

The simplest method of spacing the ellipse 
is to step about it with the dividers. If the 
curve is flat, the dividers should be set to less 
than a whole tooth, for equal chords will not 
measure equal arcs of the curve. 

But this stepping method, although it is 
sufficient and convenient for drafting pur- 
poses, is wholly unfit for mechanical pur- 
poses, and therefore we must have a method 
that is not dependent on personal skill. 



If the ellipse is drawn by means of the 
trammel, Fig. 124, it can be accurately spaced 
by means of a graduated index circle /, hav- 
ing a diameter equal to the sum of the diam- 
eters of the ellipse, for then the center line of 
the bar will pass over an arc on the ellipse 
that at the apices is exactly equal to half the 
arc passed over at the same time on the circle, 




Fig. 128. 



and that is elsewhere very nearly in the same 
proportion. 

This method is not mathematically exact, 
but its accuracy is very far within the re- 
quirements of practice. The space on the 
quarter, at Q, will be greater than anywhere 
else, but the maximum error will in general 
be very minute. 

For an example, take an extreme practical 
case, a gear with axes, eight and ten inches 
long, and with seventy-two teeth. The max 
imum error, the difference between the long- 
est and shortest tooth arcs, will be not over 
one five-hundredth of an inch. In the more 



Involute Elliptic Gears. 



i i 



common practical case of a gear of nine and 
ten inches axes, and seventy-two teeth, the 



readily calculated, but will he about one 
twenty -thousandth of an inch. In all cases 



maximum error is about one two-thousandth , likely to be met with in practice, the inevit- 
of an inch. In both these cases, the differ- able mechanical errors are greater than the 
ence between the tooth arc at the major apex j theoretical errors of the method, and it is 
and that at the minor apex is too small to be i serviceable on ellipses as flat as three to one. 




Involute elliptic teeth* 

Fig, 129. 



143. — INVOLUTE ELLIPTIC TEETH. 



As in the case of the circular gear, the 
best form of tooth for the elliptic gear is the 
involute, and for the same reasons. 

The base line of the involute tooth is any 
ellipse BE, Fig. 125, which is drawn from 
the same foci as the pitch ellipse ; the limit 
point i is the point of tangency of a tangent 
from the pitch point 0, and the addendum 
line a I of the mating gear must not pass 
beyond that point. The method of laying 
out the tooth and drafting it is so exactly 
like the process for the circular gear that 
the description need not be repeated. 

The centers of involute elliptic gears can 
be adjusted without affecting the perfection 
of the motion transmitted, but, as the focal 



distance remains fixed, the ratio of the axes 
will be altered. 

The work of drawing the teeth can be 
much abbreviated by the process illustrated 
by Fig. 129. Find the centers for approxi- 
mate circular arcs, preferably by the method 
of (140), and then consider the gear as made 
up of four circular toothed segments. It is 
then necessary to construct but two tooth 
curves, one for the major and one for the 
minor segment, and the flanks will be radii 
of the circular arcs. 

The line of action, la, Fig. 125, is not a 
straight line, and it is not the same for all 
the teeth. It is not fixed when the pitch 
point and the line of centers is fixed (134). 



78 



Cycloidal Elliptic Gears. 




Cycloidal elliptic teeth* 

Fig. 130. 



144. — CYCLOIDAL ELLIPTIC TEETH. 



The cycloidal tooth is drawn, exactly as 
upon a circular pitch line, by a tracing point 
in a circle that is rolled on both sides of the 
pitch line. The line of action is not a circle, 
and it is not the same curve for all the teeth. 

That the flanks shall not be under-curved, 
the diameter of the rolling circle should not 
be greater than the radius of curvature at 



the tooth being drawn, and when, as usual, 
the same roller is used for all the teeth, its 
diameter should not be greater than the 
radius of curvature at the major apex, the 
distance Ch of Fig. 122. 

Fig. 130 shows a cycloidal gear drawn as 
four circular segments, by the methods of 
(140) and (83). 



145. — IRREGULAR TEETH. 



It is most convenient to draw all the teeth 
alike, with the same rolling circle, or from 
the same base line, and also to uniformly 
space the pitch line, but such uniformity is 
not essential. 



The only requirement is that each tooth 
curve shall be conjugate to the tooth curve 
that it works with, and if that condition is 
satisfied the teeth may be of all sorts and 
sizes. 



146. — FAILURE IN THE TOOTH ACTION. 



When the major axes are in line the action 
of the teeth on each other is nearly direct, 
but when the minor axes are in line the action 



is more oblique, as shown by Fig. 127. The 
teeth tend to jam together when the driver 
is pushing the follower, and to pull apart 



The Link. 



79 



when the follower is being pulled, and when 
the ellipse is very flat this tendency is so 
great that the teeth fail to act serviceably. 

At first glance it might appear that this 
difficulty in the tooth action of very eccentric 
gears might be overcome by making the teeth 
radial to the focus, as shown by Fig. 131, 
but examination will show that but little can 
be gained in that way. 

The teeth on the gear C were obtained by 
the method of (28) from the assumed tooth 
on the gear c, and the effect of the defective 
shape of one side of the assumed tooth was to 
cut away the con j agate curve of the derived 
tooth. 

Such teeth would not work as well as the 
ordinary form, and their construction would 
be very difficult. 




Hadial teeth 

Fig. 131. 



147. — THE LINK. 



When the teeth of the elliptic gear fail to 
properly engage, on account of the obliquity 
of the action, the difficulty can be entirely 
overcome by connecting the free foci by a 
link (141), as shown by Fig. 127. 

This link works to the best advantage 
when the teeth are working at the worst, and 
when it fails to act, as it passes the centers, 
the teeth are working at their best. There- 



fore gears that are connected by a link need 
teeth only at the major apices. 

When the tooth action is imperfect by rea- 
son of its obliquity, and the link is not avail- 
able or desirable, the difficulty can be over- 
come by using three or more gears in a train, 
as shown by Fig. 137, for then the same re- 
sult can be obtained by the use of gears that 
are much more nearly circular. 



148. — VARIABLE SPEED AND POWER. 



If the shaft c, Fig. 132, turns uniformly, 
the slowest speed of the shaft C will occur 
when the gears are in the position of the 
figure, and the proportion between the two 
speeds will be the proportion between the 
distances cO and CO. The greatest speed 
of the driven shaft will occur when the shafts 
have turned through a half revolution from 
the position of the figure, and the relative 
speed will be the same, reversed. 

The ratio of speed, the ratio of the greatest 
speed to the slowest speed, is the square of 
the ratio between the speed of the driving 
shaft and the greatest or the least speed of the 
driven shaft, so that it requires but a slight 




Fig. 132. 



variation of the axes to produce a decided 
variation of the speed. 

The following table will give the propor- 
tion of minor to major axes that will give 
any desired ratio of speeds. 



80 



, Elliptic Quick Return Motion. 



Ratio of Speeds. 

2 


Ratio of Axes. 
9b5 


3 


962 


4 


952 


5 


924 


6 


907 


7 


892 


8.. . 


878 


9 


868 


10 


854 


11 


844 


12 

13 


834 

824 


14 


817 


15 


807 


16 


800 



The power is always inversely proportional 
to the speed. If the variable shaft is running 
twice as fast as the uniform shaft, it will ex- 
ert but one-half the force. 



When the gears are arranged in a train, as 
in Fig. 137, the speed ratio for the second, 
third, and following gears will be in the pro- 
portion of the first, second, third and follow- 
ing powers of the first ratio. 

Thus, the ratio for a pair of gears with 
axes in the proportion of .952 to 1 being 4 
for the second gear, will be 16 for the third 
gear, 64 for the fourth gear, and so on. 

The use of gears of troublesome eccentric- 
ity can be avoided by this means. A train 
of three gears of .952 axes, Fig. 137, is 
equivalent to a single pair of very flat gears 
with .800 axes, Fig. 138, and, in general, 
three gears that are nearly circular are equiva- 
lent to a single very flat pair. 



149.— ELLIPTIC QUICK RETURN MOTION. 



If the gears are arranged with respect to 
the piece to be reciprocated, in the manner 
shown by Fig. 133, the time of the cutting 
stroke will be to the time of the return stroke, 
as the angle PEK is to the angle PEF, 
where E and i^are the foci of the ellipse. 

The following table will show the ratio of 
axes that must be adopted to produce a re- 
quired ratio of stroke to return. 

Quick Return. Ratio of Axes. 

2 tol 964 

3 tol 910 

4tol 861 

5 tol 817 

6 tol 778 

To determine the ellipse that will give a 
required quick return, we lay off the angles 
PEK and PEF in the given proportion, 
and then find by trial a point P such that the 
length PE plus the length of the perpendicu- 
lar PF is equal to the known center distance 
Ee. F will be the other focus of the re- 
quired ellipse. 

When the driving gear has turned through 
the angle PEF, from the position of the 
figure at the middle of the return, the varia- 
ble gear will have turned through the angle 
P"eO = P'FO, and we can study the action 
of the tool by drawing equi-distant radii 
about E, and finding the corresponding radii 
about F. 




Quick return 

Fig. 133. 



Fig. 134 shows the arrangement of the 
radii (PF = P"e of Fig. 133) in the case of a 
four to one quick return, and it is seen, by 
the parallel lines, that the motion of the tool 
is very uniform, coming quickly to its maxi- 
mum speed, and holding a quite uniform 
speed until near the end of the stroke. Fig-. 
135 shows that the same motion derived from 
a simple crank is not as uniform. 

When the gears are arranged in a train, 
Fig. 137, the quick return ratios can be de- 
termined by the construction shown by Fig. 
136. Drawee at right angles to AA', and 
draw cEd through the other focus. The 
quick return ratio of the second gear will be 
the ratio of the angles a» and b 2 . Draw 
dFe, and the ratio for the third gear will be 



Elliptic Trains. 



81 




Quick return crank 

Fig. 134. 



Ordinary crank 

Fig. 135. 



that of the angles a s and b a . Draw eEf, 
and a A and 6 4 will give the ratio for the 
fourth gear. And so on, in the same man- 
ner, as far as desired, the ratio being greatly 
increased by each gear that is added to the 
train. 

If carefully performed, the graphical pro- 
cess is quite accurate. The case of axes in 
the proportion of .98 to 1 gave a quick re- 
turn of 1.6 for the second gear, and 2.8 for 
the third gear, while their true computed 
values are 1.66 and 2.74. 

The chart will solve quick return 
train questions involving gears not 
flatter than .80, as accurately as need 
be. For example, the ratio of axes 
of .95 will give a quick return of 
2.25 for the second gear, 4.85 for 
the third gear, 9.80 for the fourth 
gear, and 19.70 for the fifth gear. Again, 
the proportion of axes to give a quick return 
of 5 for the third gear is .948. 




Quick return train 

Fig. 136. 




Elliptic train Fig. 137. 




Fig. 138. 



82 



Elliptic Gear Cutting Machine. 



ft 

c 
© 



Elliptic Quick Return Chart- 





































































































































































































































W> 






^ 






































-& 
































Y 


s& 




v^ 
























jy^ 










v©. 




























&? 






































<e 








r 


\ 




V 
















































































































































































Si 


§fifl»i 


d-£ 






































?ar~ 










































































Fir 


st gear 


































1 

























.60 ,81 .82 .83 .84 .85 .86 .87, .88 ,89 .90 .91 .92 <93 34 .95 .96 .97 

Proportion of JLxes 



.99 1.00 



150. — THE ELLIPTIC GEAK CUTTING 
MACHINE. 

The conditions of the described 
operation of drawing the ellipse by 
means of the trammel (138) may be 
reversed, the bar being held still 
while the paper and the cross are 
revolved, and it is evident that the 
result will be the same ellipse on the 
paper as if the bar is revolved as 
described. 

By thus reversing the process of 
describing the ellipse, and by adopt- 
ing the improved spacing device of 
(142), we can construct a machine 
for accurately cutting the teeth in 
an elliptic gear, the main features 
of which, omitting various unessen- 
tial details, are shown by Figs. 139 
and 140. 

The blank to be cut is fastened 
upon a trammel stand, which cor- 
responds to the paper in the graphi- 
cal process, and revolves upon the 
fixed base. The adjustable trammel 
pins a and b are fixed in a slot in 
the bed, and they fit and slide in the 
slots M and N in the under surface 
of the stand. The cutter which 
corresponds to the tracing point is 
fixed with the pitch center of its 




Plan 
Fig. 139. 



Cutter 




Elevation 
Fig. 140. 



Elliptic Bevel Gear. 



83 



tooth curve directly over the point P in the 
line of the pins. The index plate has a 
diameter equal to the sum of the axes of the 
ellipse, and it is held by an index pin p, 
which slides in the slot, and is always'in the 
liDe of the pins. 

Thus arranged, the machine will always 
cut its tooth in the true ellipse, and the teeth 
will be accurately spaced. 

The direction of the tooth will be sub- 
stantially at right angles to the pitch line, 
and a simple arrangement can be applied to 
make it exactly so. An index plate of a 
fixed diameter may be used for all sizes of 



gears, if the index pin is carried by an arm 
which swings about the center of the gear, 
and has an adjustable pin that slides in the 
slot. 

The tops of the teeth are trued by a cutter 
having a square edge, and the line of the 
tops will be substantially parallel to the pitch 
line. 

The blank is held by an arbor through its 
focus hole, and the arbor is held by a slide, 
which slides in a chuck upon the stand, so 
that the focus can be accurately set in the 
major axis at the proper distance from the 
center. 



151. — CHOICE OF CUTTERS. 



Theoretically, the teeth are of different 
shapes, as they are in different positions upon 
the ellipse, and, therefore, each space should 
be cut with a cutter that is shaped for that 
particular space. But as this is impracticable, 
it is necessary to choose the cutter that will 
serve the best on the average. 

Strictly, the cutter should be the one that 
is fitted to cut a spur gear having a pitch 
radius equal to the radius of curvature of 



the ellipse at the major apex, but as that 
cutter will be much too rounding for the 
minor apex, it is better to choose the one 
that is fitted for the medium radius of cur- 
vature. 

The two radii of curvature are the dis- 
tances Ch and Gk, Fig. 122, and the cutter 
should be chosen for the radius half way 
between the two, approximately half the 
sum of the two. 



152. — THE ELLIPTIC BEVEL GEAR. 



An ellipse may be drawn on the surface of 
a sphere by means of a string and two pins, 
according to the method of (138), and a 
pair of such spherical ellipses will roll on 
each other while fixed on their foci, their 
free foci moving at a constant distance 
apart. 

Therefore we can have elliptic bevel 
gears that are very similar to elliptic spur 
gears, as shown by Fig. 141. The two gears 
revolve on radial shafts through their foci, 
and the link connects radial shafts through 
the free foci. The velocity ratio is the ratio 
of the perpendiculars a b and a c. The 
elliptic bevel gear is the invention of Pro- 
fessor MacCord. 

The spherical ellipse cannot be drawn by 



the trammel method of (138), and therefore 
the method of spacing of (142), as well as 




Elliptic bevel gea'rs 

Fig. 141. 

the gear cutting machine of (150), does not 
apply. 



84 



Elliptic Calculations. 



153. — MATHEMATICAL TKEATMENT. 



If the major semi-axis is a, and the minor 
semi- axis is b, the equation of the curve from 
the origin at C is 

a 2 y% _j_£3 x z — a % 5^ 

the major axis being the axis of X. 

The distance OF from the center to the 
focus will be 

c = *J a* — b 2 = a Vl— n 2 , 

in which n is the ratio of axes = — . 

a 

The radius of curvature at the major apex 

7,2 z.8 

is — , and that at the minor apex is _ . 
a b 

There is no practicable formula for the recti- 
fication of the curve, as the length is express- 
ible only by a series. 

The special spacing method of (142) is true 
only at the instant of passing either apex, 
for the tracing point describes half the arc 
described by the line of the bar on the index 
circle only when the bar is at right angles 
with the curve. The error will be at its 
maximum when the bar is at the maximum 
angle with the normal, which is at about an 
angle of forty-five degrees with the major 
axis. The difference between an ordinary 
tooth space at the major apex, and that 
at the minor apex, is very minute. A very 
careful calculation of the length of the chord 
of a gear of seventy-two teeth, and eight and 
ten inch axes, gave a chord of .41433" at the 
major apex, a chord of .41495" at 45° for the 
maximum, and a chord of .41441" at the 
minor apex. The difference between the 
chords at the apices is .00008", but as the cur- 



vature at the major apex is greater than at the 
minor apex, the difference between the arcs 
would be less, perhaps not over .00004". 
The ratio of speeds (148), is 



(i 



+ \A- 



n x 



VI — n 2 J 
The ratio of quick return being given as 
qr, the value of n is 



n 



= V 2 V d* + d 4 — 2d 9 , 
180 



in which d — tan. 



i 180 y 
[v + i)' 



When the gears are in a train, there seems 
to be no simple method for computing the 
ratio of axes to produce a given quick, 
return, but, when the ratio is given, the 
quick return for each gear can be computed 
best by trial and error with the formula 
sin. (M — JV) 



= VT 



n< 



sin. if -|~ sin. iV 

in which M is any known angle b, Fig. 136, 
and -ZVis the angle b for the next following 
gear in the train. Thus, assuming n = .98, 
and M t = 90°, we find JV & = 67° 28'. Then 
putting M 2 = 67° 28', we find JV 8 = 48° 5'. 
Knowing the angles, we compute the 
quick return ratio from 

— ?-* 

which, for n = .98 gives qr for two gears 
equal to 1.66, and for three gears equal to 
2.74. The graphical process of Fig. 136 
should first be employed to fix the angles 
approximately. 



8. THE BEVEL GEAR. 



154. — THE BEVEL GEAR. 



The theory of the bevel gear cannot be 
properly represented, and can be studied 
only with the greatest difficulty, upon a 
plane surface. It is essentially spherical in 
nature, and should be- shown upon a spheri- 
cal surface, as in Figs. 143 and 144. 

This is best dpne upon a spherometer, 
which is simply a painted sphere fitted in a 
ring. The sphere rests upon a support, so 
that the ring coincides with a great circle 
upon it, and the ring is graduated to 360°. 
A very roughly made wooden sphere and 
plain ring will be found to answer the gen- 



eral purpose very well, and should be pro- 
vided if the study of the bevel gear is 
seriously intended. If painted, ink marks 
can be scrubbed off, and pencil marks re- 
moved with a rubber. 

The mathematical treatment is unapproach- 
able without a knowledge of the common 
principles of spherical trigonometry. 

A wide, interesting, and difficult field of 
study is offered, but space will permit but a 
brief examination of the more prominent 
and practical points. A careful examination 
would require ten times the available space. 



155.— THE GENERAL THEORY. 



When thus represented upon the spherical 
surface, the theory of the bevel gear is so 
similar to that of the spur gear, as repre- 
sented upon a plane surface, that any de- 
tailed description would be mostly a repeti- 
tion of what has already been stated. 

All straight lines of the spur theory are 
represented by great circles, the crown gear 
being the rack among bevel gears, and all 
distances are measured in degrees. 

Irregular pitch lines and multilobes are 
managed substantially as for spur gearing. 
The elliptic bevel gear has been described in 
connection with elliptic spur gears (152). 

The tooth surfaces of the bevel gear arc 
generally formed by drawing straight lines 
from the spherical outline to the center of the 
sphere, as in Figs. 143 and 144, the pitch lines 
and tooth outlines being the bases of cones 
with a common apex. 

When limited in width, as is usually the 
case, it is by a sphere concentric with the 
outside sphere, so that a spherical shell is 
formed. 

These concentric spherical shells can be 
moved on their axes to form twisted and 
spiral teeth, Fig. 142, precisely as described 
for spur gears (99). 



The molding process of (27) will apply per- 
fectly, but it has but one practical applica- 
tion. 

Fig. 142. 




Twisted bevel gear. 



The planing process of (28) will fail, for 
practical purposes, except for one particular 
form of tooth, because the shape of the cut- 
ting tool cannot in the general case be 



86 



Involute Bevel Gears. 



changed in form as it approaches the apex, 
and therefore the tooth will not be conical. 

The planing process of (29) will apply per- 
fectly, the strokes of the tool being radial, 
and on this method we must depend for the 
accurate construction of all forms of bevel 
gear teeth except the octoid and the pin 
tooth. 

As the diameter of the sphere is increased, 
the radii become more nearly parallel, until, 
when the diameter is infinite, they are paral- 



lel. Therefore the spur gear is a particular 
case of the bevel gear, and all formulae and 
processes that apply to the bevel gear will 
apply to the spur gear if the diameter of the 
sphere is made infinite. The most scientific 
method of study would be to develop the 
theory of the bevel gear, and fiiom that pro- 
ceed to that of the spur gear, but such a 
method would be difficult to clearly carry 
out, and is best abandoned for the more con- 
fined process here adopted . 



156. — PARTICULAR FORMS OF BEVEL TEETH. 



As in the case of spur gearing, there can 
be an infinite number of tooth curves for 
bevel gearing (31), each form having its own 
line of action, but as there are only four 
forms that are available for practical use by 
means of simple processes of construction, 
our attention will be confined to them. 

These four particular forms are, first, the 



involute tooth, having a great circle line of 
action; second, the cycloidal tooth, having a 
circular line of action; third, the octoid 
tooth, having a plane crown tooth, and a 
"figure eight" line of action; and, fourth, 
the pin tooth, for which one gear of a pair 
has teeth in the form of round pins. 



157. — THE INVOLUTE BEVEL TOOTH. 



The spherical involute must be studied as 
a whole if its form is to be clearly seen. 

Its definition is that it is the tooth curve 
having a great circle for a line of action. In 
Fig. 143 the great circle line of action la ex- 
tends around the sphere at an angle with the 
crown pitch line pi, and it is tangent to 
two base lines bl and bl', that are paral- 
lel with the crown line. 

The most convenient method of draw- 
ing the tooth curve is by rolling the line 
of action on the base line, while a point 
in it describes the curve on the surface 
of the sphere. The equivalent graphical 
process is to step along the base line 
and any two tangent great circles, from 
any point on the curve to any desired 
point. 

It will take the form shown by the 
dotted lines ; rising at right angles to the 
base line, it curves until the crown 
line is reached, there reversing its curva- 
ture and bending the other way until it 
meets the other base line. At the base 
line it has a cusp, and rises from it to 
repeat the same course indefinitely. 



Fig. 143 shows a crown gear or rack. The 
pitch line is the great circle pi. The line 
of centers cO C is a great circle at right angles 
with the crown line pi. The line of action 
is the great circle la set at a given angle of 
obliquity with the crown line. The base 

Fig. 143. 

Xlie involute Tooth, 




Bilgravi Bevel Gears. 



87 



circles are the small circles bl and bl'. The 
spherical involutes have the same property of 
adjustability as have the spur involutes, the 



motion being confined to the sphere, and there- 
fore the gears are adjustable as to their shaft 
angle, the apex remaining common to both. 



158. — THE CYCLOIDAL BEVEL TOOTH. 



The definition of the cycloidal tooth is 
that it is that form which has a circular line 
of action. 

The rolled curve method of treatment (32) 
applies, and is the best means of studying 
the curve. 

There is no gear with radial flanks, the 



flank formed by a roller of half the angular 
diameter of the gear being nearly but not 
exactly a plane. 

The theory differs so little from that of 
the spur gear, that but little of interest can 
be found, and the curve will not be consid- 
ered further. 



159. — THE OCTOID BEVEL TOOTH. 



The definition of this tooth system is that 
it is the conjugate system derived from the 
crown gear having great circle odontoids. 

In Fig. 144 the crown gear has plane 
teeth cutting the sphere in great circles, 
mOn, while a pinion would have convex 
tooth curves conjugate to the great cir- 
cles of the crown tooth. 

The line of action, from which the 
tooth derives its name, is the peculiar 
' ' figure eight " curve la, which is at right 
angles to the tooth curve at the crown 
line pi, and tangent to the polar circles 
S and 8', to which the great circle crown 
odontoids are also tangent. 

This tooth owes its existence to the 
fact that it is the only known tooth, and 
probably the only possible tooth, that 
can be practically formed by the mold- 
ing planing process of (28). The cutting 
edge of the tool being straight, no 
change is required while it is in motion, 
except in its position, and that is accom- 
plished by giving it a motion in such 
a direction that its corner moves in the radial 
line of the corner of the bottom of the tooth 
space. 

The octoid tooth, together with an ingeni- 



ous machine for planing it, was invented by 
Hugo Bilgram, but it has always been mis- 
taken for the very similar true involute tooth. 




Tlie-Octoid Tooth 

Fig. 144. 

Bilgram's machine is described in the 
American Machinist for May 9th, 1885, 
and in the Journal of the Franklin Institute 
for August, 1886. 



160. — THE PIN BEVEL TOOTH. 

If the tooth of one gear of a pair is a coni- j gate to it, and the combination deserves 
cal pin, Fig. 145, with apex at the center of notice because it is one of the few forms that 
the sphere, that of the other will be conju- 1 are easily constructed. It may be said that 



88 



Tredgold s Method. 



its practical construction is simpler and easier 
than that of any other form of bevel gear 
tooth except the skew pin tooth of (180). 

Fig. 145. The tooth is 

preferably, but 
not necessarily, 
of the conical 
form, for other 
forms of circu- 
lar pins would 
serve the theo- 
retical pur- 
pose. 
Pin bevel gears^ Its theory is, 

in the main, the same as that of the spur pin 




tooth. It has the same troublesome cusp, 
which can be avoided in the same way, by 
setting the center of the pin back from the 
pitch line. 

It is the only known form of tooth that 
can be formed in a practical manner by the 
molding process of (27). If the cutting tool 
is a conical mill, it will form the conjugate 
tooth while the two pitch wheels are rolled 
together. 

The pins may be mounted on bearings at 
their ends, forming roller teeth. They 
would be weak, but would run with the 
least possible friction, all the rubbing friction 
being confined to the bearings. 



161. — tredgold's approximation. 



The construction of the true bevel gear 
tooth curve upon the true spherical surface is 
impracticable with the«means in ordinary use, 
and the true method of computation by means 
of spherical trigonometry is equally unfitted 
for common use. But, by adopting Tred- 
gold's approximate method the difficulties can 
be overcome. 

By this method the tooth curves are drawn, 
not on the true spherical surface, but, as in 
Fig. 146, on cones A and B drawn tangent 
to the sphere at the pitch lines of the gears. 
The cones are then rolled out on a plane sur- 
face, and the gear teeth drawn upon them 
precisely as for spur gears of the same pitch 
diameter. 

Practically correct tooth curves could thus 
be drawn on the spherical surface by cutting 
the teeth to shape, and bending them down 
to scribe around them, but in practice the 
back rims of the gears are shaped to the tan- 
gent cones so that the teeth lie directly upon 
the conical surface. 

This method is called approximate, but its 
real error would be difficult to determine, 
and is certainly not as great as the inevitable 
errors of workmanship of any graphical pro- 
cess. The tooth outline drawn by it upon 
the spherical surface may be considerably 
different from that which would be drawn 
directly upon it, but it does not follow that 
it is therefore incorrect. The only require- 
ment is that the engaging curves shall be 



Fig. 146 




Tredgold's method, 

conjugate odontoid s, and it is a matter of 
very small consequence whether or not the 
curve on the sphere is the same kind of curve 
as that upon the cone. If the true plane in- 
volute curve is drawn upon the developed 
cone, the corresponding curve on the sphere 
will not be an exact spherical involute, but its 
divergence from some true odontoidal shape 
must be minute, even when the teeth are very 
large indeed. In ordinary cases it cannot be 
sufficient to affect materially the constancy 
of the velocity ratio. What is sometimes 
given as its error is mostly the ' ' difference in 
shape " between the plane and the spherical 
teeth. 



Drafting Bevel Gears. 



89 



162. — DRAFTING THE BEVEL GEAR. 



The practical application of Tred gold's 
method is illustrated by Fig. 147. 

Draw the axes GA and GB at the given 
shaft angle AGB. Lay off the given pitch 
radii a and b, and draw the lines c and d in- 
tersecting at the pitch point 0. Dra the 
center line 0(7, and lay off the face Of. 

The pitch diameters are ON and OM, 
and NGO and MCO are the pitch cones. 

Draw the back rim line 01) at right an- 
gles with the center line, lay off the addenda 
Oe and Og, and the clearance gh. Draw the 
front rim line parallel to the back rim line. 



The center angle is X, the face increment 
is F, and W is the face angle. The cutting 
decrement is J, and T is the cutting angle. 
Twice the distance ran is the diameter incre- 
ment, and em is the outside diameter. 

The pitch radius of the Tredgold back 
cone is OB, and the figure shows the con- 
struction of the gear teeth on this cone 
developed. The teeth are represented as 
drawn upon the figure, but it is better to use 
a separate sheet. The odontograph should 
be used, calculating the number of teeth in 
the full circle of the developed cone. 



Fig. 147. 




Drafting the bevel gear.* 



163. — THE BEVEL GEAR CHART. 



The drafting of the bevel gear blanks by 
means of the method of (162) is simple, but 
the method requires drafting instruments, 
not always at hand, as well as the ability to 
use them accurately. The drawing must be 
carefully made, to give correct results, par- 
ticularly when the gears are small. After 
the drawing is made the various angles and 



diameters must be taken off for use at the 
lathe, and that is by no means a simple mat- 
ter. 

So great are the practical difficulties that 
any one who has a knowledge of simple arith- 
metic will find it not only easier, but more 
accurate to use the chart and method by 
means of the following rules. 



90 



Bevel Gear Chart. 







THE 


BEVEL GEAR CHART. 






Shafts at 90° 
Proportion. 


Center 
Angle. 


Angle Incr. 
Div. by Teeth 


Diam. Incr. 
Div. by P. 


Shafts at 90° 
Proportion. 


Center 
Angle. 


Angle Incr. 
Div. by Teeth 


Diam. Incr. 
Div. by P. 


.10 


1—10 


5.71 


11 


2.00 


10.00 


10—1 


84.29 


114 


.20 


.11 


1— 9 


6.28 


13 


2.00 


9.00 


9—1 


83.66 


114 


.22 


.13 


1— 8 


7.41 


14 


1.99 


8.00 


8—1 


82.87 


113 


.25 


.14 


1— 7 


7.97 


16 


1.98 


7.00 


7—1 


81.87 


113 


.28 


.17 


1— 6 


9.65 


19 


1.97 


6.00 


6—1 


80.54 


113 


.33 


.20 


1— 5 


11.81 


23 


1.96 


5.00 


5—1 


78.69 


112 


.39 


.22 


2— 9 


12.40 


25 


1.95 


4.50 


9—2 


77.47 


111 


.43 


.25 


1— 4 


14.04 


28 


1.94 


4.00 


4—1 


75 97 


111 


.49 


.29 


2— 7 


16.17 


32 


1.92 


3.50 


7—2 


74.06 


110 


.55 


.30 


3—10 


16.70 


33 


1.92 


3.33 


10—3 


73.28 


109 


.58 


.33 


1— 3 


18.20 


36 


1.90 


3.00 


3—1 


71.57 


109 


.63 


.38 


3— 8 


20.80 


40 


1.87 


2 67 


8—3 


69.47 


107 


.70 


.40 


2— 5 


21.80 


43 


1.86 


2.50 


5—2 


68.19 


106 


.74 


.43 


3— 7 


23.27 


45 


1.84 


2.33 


7—3 


66.78 


105 


.79 


.44 


4— 9 


23.75 


46 


1.83 


2.25 


9—4 


66.04 


104 


.81 


.50 


1— 2 


26.57 


51 


1.79 


2.00 


2—1 


63.44 


103 


.89 


.56 


5— 9 


29.25 


56 


1.74 


1.80 


9—5 


60.95 


101 


.97 


.57 


4- 7 


29.67 


57 


1.74 


1.75 


7—4 


60.25 


99 


.99 


.60 


3— 5 


30.97 


59 


1 72 


1.67 


5—3 


59.08 


98 


1.03 


.63 


5— 8 


32.20 


61 


1.69 


1.60 


8—5 


58.00 


97 


1.06 


.67 


2— 3 


33.82 


64 


1.66 


1.50 


3—2 


56.31 


95 


1.11 


.70 


7—10 


34.99 


66 


1.64 


1.43 


10—7 


55.03 


94 


1.15 


.71 


5— 7 


35.38 


67 


1.63 


1.40 


7—5 


54.46 


93 


1.16 


.75 


3— 4 


36.87 


69 


1.60 


1.33 


4—3 


53.07 


92 


1.20 


.78 


7— 9 


37.95 


70 


1.58 


1.29 


9—7 


52.22 


91 


1.22 


.80 


4— 5 


38.66 


72 


1.56 


1.25 


5—4 


51.34 


90 


1.25 


.83 


5— 6 


39.69 


73 


1.54 


1.20 


6—5 


50.20 


88 


1.28 


.86 


6— 7 


40.70 


75 


1.52 


1.17 


7—6 


49.48 


87 


1.31 


.88 


7— 8 


41.35 


76 


1.50 


1.14 


8—7 


48.75 


86 


1 32 


.89 


8— 9 


41.67 


76 


1.49 


1.13 


9—8 


48.49 


86 


1.33 


.90 


9—10 


41.98 


77 


1.49 


1.11 


10—9 


47.98 


85 


1.34 


1.00 


1— 1 


45.00 


81 


1.41 


1.00 


1—1 


45.00 


81 


1.41 



Specimen Chart Calculations. 



91 



Fig. 148. 
Sample Computation. 



SHAFTS AT 
A BIGHT ANGLE. 



Pitch = 3 Prop. = 7 — 5 


Shaft ang. 90 


Teeth = 42) 93 (2.22 = 
84 .37 + 

90 2.59 = 

84 

60 


face incr. 

i 
s 

cut deer. 


Center angles = 54.47 
-finer 2.22 


35.38 
2.22 




Face angles 56.69 


37.60 


Center angles = 54.47 
— deer 2 59 


35.38 
2.59 


Cut angles , . . 51.88 


32.79 


Pitch = 3) 1.16 

Diam. incr. = .39 
-{- p. diams. = 14. 


3) 1.63 

.54 
10. 


o. diams. = 14.39 


10.54 



Fig. 149. 
Sample Computation. 



SHAFTS AT 
ANY ANGLE. 



1 

Pitch = 5 Prop. = X 


Shaft ang. 52.8 


Teeth = 20) 66 (3.30 
.55 

3.85 


— face incr. 

+ 1 

= cut deer. 


Center angles = 35.80 
-f incr 3.30 


17.00 
3.30 




Face angles = 39.10 


20.30 


Center angles = 35.80 
— deer 3.85 


17.00 
3.85 


Cut angles = 31.95 


13.15 


Pitch = 5) 1.66 


5) 1.91 


-{- p. diams 4. 


.38 
2. 


o. diams 4.33 


2.38 



92 



Chords of Angles. 



164. — SHAFTS AT EIGHT ANGLES. 



1st. — Divide the pitch diameter by that of 
the other gear of the pair, or else the number 
of teeth by that of the other gear, to get the 
proportion. Enter the table by means of the 
proportion. All numbers for that pair will 
be found on the same horizontal line in the 
two columns. 

2d. — The center angles are given directly 
by the table at the proper proportion. 

3d. — Divide the tabular angle increment 
by the number of teeth in the gear, to get 
the angle increment. This need be done for 
but one gear of a pair, as the increment 
is the same for both. 



4th. — Add the angle increment to the cen- 
ter angle, to get the face angle. 

5th. —Increase the angle increment by one- 
sixth of itself, to get the cutting decrement, 
and subtract this decrement from the center 
angle, to get the cutting angle. 

6th. — Divide the tabular diameter incre- 
ment by the diametral pitch, to get the 
diameter increment, and add that to the pitch 
diameter, to get the outside diameter. 

Fig. 148 is a sample computation for shafts 
at right angles. 



165.— SHAFTS NOT AT BIGHT ANGLES. 



The table cannot be entered by means of 
the proportion, and the numbers for the two 
gears of the pair will not be found on the same 
horizontal line, and it will be necessary to 
determine the center angles. 

As in Fig. 147, draw the axes, at the given 
shaft angle, and find the center angles, by the 
method described in (162). 

Then enter the table, for each gear by 
itself, by means of the center angles, and 
proceed as for shafts at right angles. The 
angle increment and decrement is the same 
for both gears of a pair. 

Fig. 149 is a sample computation applied 



to the case of Fig. 147, the center angles be- 
ing found by means of the table of chords. 
If preferred, the center angles can be found 
by means of the formula, 

sin. 8 



tan. C = 



— 4- cos. 
P^ 



S 



in which G is the center angle of the gear, P 
is the proportion found by dividing the num- 
ber of the teeth in the gear by the number in 
the other gear, and 8 is the shaft angle. 
Having found one center angle, subtract it 
from the shaft angle to get the other center 
angle. 



166. — THE TABLE OF CHOBDS AT SIX INCHES. 



When the lathesman is provided with a 
graduated compound rest which feeds the 
tool at any angle, nothing but the computa- 
tion is required; but when there is nothing 
but the common square feed, the faces must 
be scraped with a broad tool. A templet for 
guiding the work can easily be made by 
means of the table of chords at six inches. 

To lay out a given angle, draw an arc 
with a radius of six inches, draw a chord of 
the length given by the table for the angle, 



and then draw the sides oc and db of the 
angle hoc, Fig. 150. 

For tenths of a degree use the small tables. 
The chord of 37.5° is 3.81 + .05 = 3.86 
inches. 

Fig. 151 shows the manner of using ihe 
angle templet at the lathe. 

This table of chords is very convenient for 
many purposes not connected with gearinr, 
and it is more accurate tnan the common 
horn or paper protractor. 



167. — bilgbam's chabt. 
A graphical method for determining the 



angle and aiameter increments, the invention 
of Hugo Bilgram, is described in the Amebi- 
can Machinist for November 10, 1883. It 



determines the required values by the inter- 
sections of lines and circles, and requires no 
computation. 



Chords of Angles. 



93 




Chord of an angle 

Fig. 150. 




Using the templet 



TABLE OF CHORDS OF ANGLES, 

AT RADIUS OF SIX INCHES. 



Degrees 
1 


Chord. 


Tenths. 


Degrees 


Chord. 


Tenths. 


Degrees 


Chord. 


Tenths. 


.10 




31 


3.20 




61 


6.10 




2 


.20 




32 


3.31 




62 


6.19 




3 


.31 




33 


3.41 




63 


6.28 




4 


.42 




34 


3.51 




64 


6.36 




5 


.52 




35 


3.61 




65 


6.45 




6 


.62 


36 


3.71 


66 


6.54 


7 


.73 




37 


3.81 




67 


6.62 




8 


.84 




38 


3.91 




68 


6.71 




9 


.94 




39 


4.01 




69 


6.80 




10 
11 


1.04 
1.15 




40 


4.10 




70 


6.89 




41 


4.20 


71 


6.97 


12 


1.26 




42 


4.30 




72 


7.06 




13 


1.36 


.1— .01 
.2— .02 
.3— .03 
.4-.04 


43 


4.40 


.1— .01 
.2— .02 

.3-.03 
.4— .04 


73 


7.14 


.1— .01 

.2— .02 
.3— .02 
.4-. 03 


14 
15 


146 
1.57 


44 

45 

46 


4.50 
4.60 
4.69 


74 
75 


7.22 
7.31 


16 


1.67 


76 


7.39 


17 


1.77 


.5— .05 


47 


4.79 


.5-. 05 


77 


7.47 


.5— .04 


18 


1.87 


.0— .06 


48 


4.88 


.6— .05 


78 


7.55 


.6— .05 


19 


1.98 


.7— .07 


49 


4.98 


.7— .06 


79 


7.63 


.7— .06 


20 


2.08 


.8— .08 
.9— .09 


50 


5.08 


.8— .07 
.9— .08 


80 


' 7.71 


.8—06 
.9-.07 


21 


2.18 


51 


5.17 


81 


7.79 


22 


2.29 




52 


5.26 




82 


7.87 




23 


2.39 




53 


5.35 




83 


7.95 




24 


2.49 




54 


5.45 




84 


8.03 




25 


2.59 




55 


5.54 




85 


8.11 




26 


2.70 


56 


5.63 


86 


8.18 


27 


2.80 




57 


5.72 




87 


8 26 




23 


2.90 




58 


5.82 




88 


8.34 




29 


3.00 




59 


5.91 




89 


8.41 




30 


3.10 




60 


6.00 




90 


8.48 





94 



Templet Planer. 



168. — KOTAKT CUT BEVEL TEETH. 



The most common method of forming the 
teeth of the bevel gear is by cutting them 
from the solid, blank by the use of the com- 
mon rotary cutter. 

The cutter should be shaped to cut the 
tooth of the correct shape at the large end, and 
the small end must be shaped either by an- 
other cut with a different cutter, or with a file. 

It is impossible to cut the tooth correctly 
at both ends, for the simple reason that the 



shape of the tooth changes, while that of the 
cutter is invariable. Therefore the result 
must always be an approximation depending 
upon the personal skill and experience of the 
workman. It is a too common practice to 
make the teeth fit at the large ends, and to 
increase the depth of the tooth toward the 
point, so that the teeth will pass without 
filing, but such teeth can be in working con- 
tact only at the large ends. 



169. — THE TEMPLET GEAR PLANEK. 



The most common method of planing the 
teeth of bevel gears is by means of devices 
adapted to guide the tool by a templet that 
has previously been shaped, as nearly as 
may be, to the true curve. The arm that 
carries the tool is hung by a universal joint 
at the apex of the gear, so that all of its 
strokes are radial, and a finger placed in the 
line of the stroke of the cutting point of the 
tool is held against the templet. There are 
many different arrangements for the purpose, 
but they are all founded on the same princi- 
ples, and differ only as to details. 



The invention of the templet gear planer is 
commonly credited to George H. Corliss, 
who patented it in 1849, and was the first to 
use it in this country. But it was patented 
in France, by Grlavet, in 1829, and may be 
even older. 

It is largely used for planing the teeth of 
heavy mill gearing, but has not been, and 
cannot be, profitably applied to common 
small gear work. Its product is, in any case, 
superior to the rough cast tooth, but its accu- 
racy is dependent on that of the templet, and 
is therefore dependent on personal skill. 



9. THE SKEW BEVEL GEAR. 



170. — THE SKEW BEVEL GEAR. 



When a pair of shafts are not parallel, and 
do not intersect, they are said to be askew 
with each other, and they may be connected 
by a pair of skew bevel gears, having 
straight teeth, which bear on each other 
along a straight line. Such gears are to be 
carefully distinguished from spiral gears, 
used for the same purpose but having spiral 
teeth bearing on each other at a single point 
only. 

We will endeavor to describe the skew 
bevel gear so that its general nature can be 
understood, but it is impossible to do so in 
simple language. It is the most difficult ob- 
ject connected with the subject. The theory 
cannot even be considered as yet settled, for 
writers upon theoretical mechanism do not 
agree upon it, and there are points yet in 
controversy. 

In the theory of the bevel gear the surface 
of reference is the spherical surface upon 
which the tooth outlines are drawn, and upon 



which the laws of their action may be 
studied, for spheres of reference of two sep- 
arate gears may be made to coincide so that 
the lines upon one will come in contact with 
those upon the other. For the spur gear, the 
spheres become planes and the process is the 
same. But for the skew bevel gear there is 
no analogous process, for it is impossible to 
imagine a surface of such a nature that it can 
be made to coincide with a similar surface 
when both are attached to revolving askew 
shafts. There are spiral surfaces which will 
approximately coincide, and are analogous to 
the Tredgold tangent cones of bevel gears 
(161), but any tooth action developed upon 
such approximate surfaces must, of necessity, 
be not only approximate, but also very diffi- 
cult to define and formulate. 

Of all the skew tooth surfaces that have 
been proposed, there is but one, the Olivier 
involute spiral oid, that can be proved to be 
theoretically correct. 



171. — THE HYPOID. 

The pitch surface of the skew bevel gear lines d and d ', Fig. 153, either one of which 



is the surface known as the " hyperboloid of 
revolution," and it is so intimately connected 
with the subject that it must be thor- 
oughly understood before going further. 
The clumsy name may be abbreviated to 
" hypoid." 

If a line D, Figs. 152 and 153, called 
a generatrix, is attached to a revolving 
shift A, so that it revolves with it, it 
will develop or ' ' sweep up " the hypoid 
H in the space surrounding the shaft. 
A section of the surface by any plane 
normal to the axis is a circle. The com- 
mon normal to the generatrix and the 
axis is the gorge radius G, and circular 
section through that line is ihe gorge 
circle. A section by a plane D, Fig. 152, 



is an element of the surface, and will form it 
if used as a generatrix. A section by any 




Hyjjoidal sections. 

Fig. 152. 




Hyperbolic sections. 

Fig. 153. 



parallel to the axis, at the gorge distance j other plane parallel to the axis will be a 
from the axis, will be the pair of straight ! hyperbola, to which the elements d and d' 



% 



Rolling Hypoids. 



are assymptotes, or lines which the curves 
continually approach, but reach only at in- 
finity. Fig. 153 shows at Q the hyperbolas 
cut by the plane Q of Fig. 152, and at R 
those cut by the plane R. The principal 
hyperbola H is the only one with which 
we are concerned. 

The hypoid is best studied as projected 
upon a plane parallel to the axis, as in Fig. 
154, in which A is the projection of the axis, 
d is that of the generatrix, dOA is the skew 
angle, and His the principal hyperbola. 

"When the skew angle and the gorge radius 
are given, the hyperbola is easily constructed 
by points. Any line ab is drawn normal to 
the axis and ihe gorge distance be= Gg is laid 
off from b, the distance ab is made equal to 
ec, and a is then a point on the curve. The 
curve is to be drawn through several points 
thus constructed. 




The hyperbola. 

Fig. 154. 

To draw a tangent to the curve at any 
point a, draw a line am parallel to the 
assymptote d, lay off mn equal to Gm, and 
draw the tangent an. 



172.— THE PITCH HYPOIDS. 



The utility of the hypoid as the pitch sur- 
face of the skew gear depends upon the pe- 
culiar property that any number of such 
surfaces will roll together, and drive each 
other by frictional contact with velocity ratios 
in the proportions of the sines of their skew 
angles, if their gorge radii are in the propor- 
tions of the tangents of their skew angles. 

It is required to construct a pair of rolling 
hypoids that will transmit a given velocity 
ratio between two shafts that are set at a 
given angle with each other. In Fig. 155, 
A and B are the given axes, and AGB the 
given shaft angle. The dirtctrix B is to be 
so drawn that the sines of the skew angles 
AGB and BGB are in the proportion of the 
given velocity ratio, and this is best done by 
drawing lines parallel to the axes, at distances 
from G that are in the given ratio, and drawing 
the directrix thr >agh their intersection B. 

In the figure the axes are situated one over 
the other at a distance GH called the gorge 
distance, and che directrix B is situated be- 
tween them so as to pass through the gorge 
line and divide the gorge distance into gorge 
radii, GW a,nd HW, which are in proportion 
to the tangents of the skew angles. This is 




Pitch hypoids. 

Fig. 155. 



best done by drawing cd normal to GB in 
any convenient position, laying off the gorge 
distance ce at any convenient angle with cd, 
and drawing de and gf parallel to it; cf will 
be the gorge radius G W for the axis GA, 



Rolli7ig Hyp oids. 



9' 



and/e will be the gorge radius iZTFfor the 
axis OB. 

Then, if the directrix, thus situated, is at- 
tached first to one shaft and then to the 
other, and used as a generatrix, it will sweep 
up a pair of pitch hypoids that will be in 
contact at the directrix, and which will roll 
on each other. 

They will not only roll on each other in 
contact at the directrix, hut they will also 
have a sliding motion on each other along 
that line, the two motions combining to form 
a resulting motion that must be seen to be 
understood. It is this sliding motion that 
makes all the difficulty in the construction of 
the teeth, for they must be so constructed 
as to allow it. It is also the cause of the 
great inefficiency of such teeth in action, for 
any possible form must have a lateral sliding 
motion, with the consequent friction and 
destruction. 

If we draw two diameters ran and m'n' 
through the same point Con the directrix, 
they will be the diameters of circles that will 
touch each other while revolving, and may 



be called pitch circles. If they are thin, and 
provided with teeth in the given velocity 
ratio, they will drive each other with a con- 
tact that is approximately correct, and if 
there are several pairs of such thin gears set 
so far apart that they do not interfere with 
each other, they will serve light practical 
purposes fairly well. 

If a face distance CE is laid off on the 
directrix and another pair of pitch circles 
constructed, the frustra of the hypoids in- 
cluded between the circles may be called 
pitch frustra, and they will roll together in 
contact at the directrix. 

It is to be noticed that the pitch diameters 
thus determined are not, as in spur and bevel 
gearing, in the inverse proportion of the 
velocity ratio of the axes, and therefore if 
one diameter of a pair of skew gears to have 
a given velocity ratio is given, the other must 
be constructed. When the skew angles are 
equal, the pitch diameters are equal, but 
otherwise the proportion cannot be expressed 
in simple terms, and must be determined by 
making the drawing. 



173. — THE LOCUS OF AXES. 



The rolling hypoids may be examined 
from another and most interesting point of 
view. In Fig. 156 the gorge line O is nor- 
mal, and the directrix D is parallel to the 
plane of the figure. The plane P is normal 
to the directrix, and below is a front view of 
it. On the plane P draw any straight line L 
through the directrix. From any two points 
a and b on this line draw lines A and B 
normal to the gorge line O, and they will 
be axes of pitch hypoids that will roll on 
each other in contact at the directrix. 

Axes drawn from all points of the line L 
will form a continuous surface called a " hy- 
perbolic paraboloid," which will be the locus 
of all the axes of a set of hypoids that will 
roll together collectively in contact at the 
directrix. 




The locus of axes* 

Fig. 150. 



98 



Olivier Skew Bevel Gears. 



174. — CYCLOID AL TEETH FOB SKEW GEARS. 



As any number of hypoids, on axes in the 
same locus of axes, will roll together in 
either external or internal contact at the 
directrix, it might be supposed that a tooth 
similar to the cycloidal tooth for bevel and 
spur gears might be formed by an element in 
an auxiliary hypoid X, Fig. 156, which rolls 
inside of one and outside of the other pitch 
hypoid. 

This is such a plausible supposition that it 
long passed for the truth, not only with its 
inventor, the celebrated Professor Willis, but 
with many other prominent writers, until 
shown by MacCord to be wrong. It serves 
to illustrate the confusion in which the whole 
subject has been and now is. 

The tooth surfaces which Willis supposed 
to be tangent at the generating elemeDt of 
the auxiliary hypoid really intersect at that 
line, and Fig. 157 shows a pair of such in- 
tersecting teeth. The curves of the figure 
were drawn by an instrument made for the 




Cycloidal tooth Curve* 
Fig. 157. 

purpose, and are, therefore, a better proof of 
the intersection of the surfaces than solid 
teeth would be. 

The cycloidal tooth is examined at consid- 
erable length, aDd the instrumental proof of 
its failure is given in the American Machin- 
ist for September 5th, 1889. 



175. — INVOLUTE TEETH FOR SKEW BEVEL GEARS. 



Herrmann's form of the Olivier spirrloidal 
tooth is constructed with the directrix of 
(172) as a generatrix, as follows : 

Suppose that cylinders are constructed 
upon the gorge circles of a pair of pitch 
hypoids, Fig. 158, and suppose a plane K to 
be placed between them. This plane will be 
tangent to both cylinders, and will contain 
the directrix, and if moved will move the 
cylinders as if by friction. Then imagine 
the plane to move in a direction normal to 
the directrix, and it will carry that directrix 
with it as a generatrix always parallel to its 
first position. It will sweep up the spiraloid 
tooth surfaces S 1 and S 2 imperfectly shown 
by the figure, or by Fig. 159, and they will 
be correct tooth surfaces always in tangent 
contact. 

Fig. 159 shows a full involute tooth sur- 
face or " spiraloid," and Fig. 160 is a full 
Olivier skew bevel gear. 

The particular involute skew tooth above 
described is not the only possible form, but 
it has the least possible sliding action, and is, 
therefore, the best. 




Involute tooth action 
Fig. 158. 

If the plane K has a generatrix line at any 
angles with the axes of the gears, and is 
moved in a direction at right angles with that 
line, correct tooth surfaces will be swept up. 
In fact, any two spiraloids on any two cylin- 
ders will work correctly with each other, and 
therefore any two spiral oidal gears of the 
same normal pitch will work correctly to- 
gether. 



Beetle's Gears. 



99 





A spiraloid 
Fig. 159. 



.Olivier Involute skew Bevel Gear 
FIq. 160. 



176. — Herrmann's law. 



Herrmann gives a law, and claims it to be 
universal, to the effect that the skew bevel 
tooth must be swept up by a straight line 
generatrix that is always parallel to the direc- 
trix. He mentions the Olivier tooth, and 
claims that it cannot be correct, evidently not 
understanding that Olivier's theory clearly 
includes the form he himself proposes. His 
form of tooth, claimed to be the only possi- 
ble form, is really only the best form of the 
Olivier tooth. 

"We will not undertake to state wherein 
Herrmann's law is incorrect, but that it is 
wrong is clearly shown by the most con- 



vincing of all proofs, the reduction to prac- 
tice. Beale, for the Brown & Sharpe Mfg. 
Co., has made working Olivier gears on a 
large scale, which are directly contrary to 
Herrmann's law, but which work perfectly, 
and demonstrate the truth of Olivier's theory 
in a way that admits of no question. 

Indeed, the closest possible scrutiny of 
Olivier's theory, without the aid of Beale's 
experimental work, fails to detect a flaw in 
it. Herrmann's condemnation of it is not 
based on direct consideration, but simply on 
the fact that it does not agree with his own 
law. 



177. — beale's skew bevel gears. 



Beale's gears are the same as Olivier's gears 
in general theory, but the improvement in 
practical form and application is so great that 
they may be considered a distinct invention. 

Fig. 161 is a section through one axis, and 
at right angles to the other axis of a pair of 
Beale gears. Both surfaces of the teeth are 
true Olivier spiraloids of Fig. 159, and the 
gears will run in either direction. When 
corrected for interference they are reversible, 
like spur or bevel gears. The gorge cylin- 
ders are tangent to each other, and are so cut 
away inside as to allow the teeth of the ma- 
ting gear to pass. 

The Olivier theory requires the teeth to 



vanish at the gorge, as shown by the single 
full tooth of Fig. 160, in order to pass, while 
the Beale gear is cylindrical in form as a 
whole, and passes the full tooth at the gorge, 
with action over its whole surface. The 
difference is practically very great. 

When in action a pair of uncorrected Beale 
gears must be placed as shown by Figs. 161 
to 163, and Fig. 169, with one end of each at 
the gorge, and they will not run together if 
placed at random. If either gear extends 
beyond the gorge line there is an interfer- 
ence between the involute spiraloids which 
is the same in kind as that between the in- 
volute curves of common spur gear teeth. 



100 



Beetle's Gears. 



gorge cylinder 
plane of action 



Each gear can drive in but one direction, 
depending upon the position of tne gear and 
the direction of the spiral, and if turned 
backwards the action is intermit- 
tent and practically useless. The 
gears must be placed as in Fig. 
162 for right-hand spirals, and as 
in Fig. 163 for left-hand spirals, 
and the direction of the rotation 
is shown by the arrow D, when 
the gear bearing the arrow is the 
driver. 

But, if the direction is to be re- 
versed, the gears can be arranged 
as in Fig. 162a, or as in Fig. 163a. 
This resetting is the same in effect 
as turning the gears half around, 
except that opposite sides of the 
teeth are in contact in the two 
positions of the same gears. 

If, however, the interfering parts 
of the tooth surface are removed, 
the gears will run together per- 
fectly and in either direction when 
put together at random as in Fig. 
168. 

In the cases shown by the figures, 
the spirals make the angles of forty- 
five degrees with the shafts, con- 
trary to Herrmann's law, but the 
action will be smoother, and the 
sliding of the teeth on each other 
will be less, if Herrmann's angles 
are adopted. These angles are the 
same as those made by the conical 
face of common bevel gears of the 
same proportion with the axes, and 
the best angles for the two-to-one 
proportion of figures are those of 
the line X of Fig. 162, making the 
angles 26° 34' and 63° 26' with the 
axes. 

The Olivier gear of Fig. 160 is 
perfect in theoretical action, but the 
teeth must be taken so far from the 
gorge that the obliquity of the ac- 
tion is excessive, and the arc of 
action is so limited that the teeth 
must be small. The sliding and wedging 
action is so great that the gears are practically 
useless. 



In comparison, the Beale gear is taken so 
near the gorge that it is practical and service- 
able, having large teeth and small obliquity. 




Section of Beetle's Gears- 



Fig. 162. 



Fig. 162a. 




'"•- 1 



Beale Shew Bevel Gears. 

Eight Hand Spiral at 45° 



Fig. 163. 



Fig. 163a. 




Left Hand Spiral\at 45 



The working length of each gear is as 
determined by the line L of Fig. 161, and 
the whole surface of the tooth within that 



Approximate Skew Teeth. 



101 



limit will be swept over "by the line of 
contact. If the length of each gear is equal 
to the radius of the other gear it will always 
be long enough. 

The action between two gears will be at 
the straight, equidistant, parallel lines a a, 
Figs. 161 and 162, in the plane of action 
tangent to both gorge cylinders. 

The shafts of a pair of skew bevel gears 
should be as near together as possible, just 
far enough apart to allow the shafts to pass, 
so as to avoid the excessive sliding action. 
In that case both Beale and Olivier gears 
are practically useless, the former on ac- 
count of the small size of the teeth, and the 
latter on account of the great obliquity of 
the action. 

The common bevel gear becomes the spur 
gear when the shaft angle becomes zero, but 
the analogous transformation of the skew 



bevel gear into a bevel gear by reducing the 
gorge distance to zero is not possible. 

The skew bevel gear becomes a spur gear 
if we imagine the axes to be brought parallel 
by removing the gorge to an infinite distance, 
for the spiraloids on the gorge cylinders then 
become involute surfaces on base cylinders. 
But, and it is a curious circumstance, when 
the shafts are brought parallel by imagining 
the shaft angles to become zero without 
changing the position of the gorge, the gorge 
cylinders become tangent and the gears do 
not become spur gears. 

Involute skew bevel gears do not appear 
to have any possible adjustment correspond- 
ing to the adjustment of the shaft distance 
of involute spur gears, or of the shaft angle 
of involute bevel gears, (56) and (157). 

Beale's gears are fully described in the 
American Machinist of Aug. 28th, 1890. 



178. — TWISTED SKEW TEETH. 



As no two surfaces of reference attached 
to a pair of revolving askew shafts can be 
made to coincide with each other, like the 
planes of spur gears or the spheres of bevel 
gears, the twisted or spiral tooth is impossi- 
ble, for such a tooth would not permit the 
required sliding action. 

But, if a line is drawn upon one pitch 
hypoid of a pair, a corresponding line may 
be drawn upon the other, as if the given 



line could leave an impression. Therefore 
a tooth having edge contact (100) may be 
constructed, provided the twist is such that 
one pair of lines always crosses the directrix. 
These teeth are purely imaginary, but if the 
edges are thick they will have an action upon 
each other, at a single point of contact, that 
is closely approximate to the theoretical 
action, and they will serve the general pur- 
pose, if the power carried is inconsiderable. 



179. — APPROXIMATE SKEW TEETH. 



As the true involute skew tooth is diffi- 
cult to construct, and in many cases is of 
small practical utility, and all other proposed 
forms are incorrect, it follows that we must 
depend for practical purposes mostly upon 
some approximation, provided it is not pos- 
sible to avoid the skew gear altogether. 

The blanks can be constructed by a definite 
process Construct the frustra of the pitch 
hypoids by the method of (172) and Fig. 155. 
Consider the end sections mn and pq as ends 
of a frustrum of a pitch cone, and on thi: 



pitch cone construct the blank gear exactly 
as for a common bevel gear. 

Having constructed the blanks, the general 
direction of the tooth is to be marked upon 
them. Mount each blank as in Fig. 155, 
with its axis parallel with a plane surface Z. 
Set a surface gauge with its point at ihe line 
of the directrix W, and with it mark the po- 
sition of the directrix on the pitch line at 
each end of the blank. 

The tooth must then be cut so that its 
Jirection follows the directrix, and it is to be 



102 



Substitute Skew Trains. 



noticed that it is not only askew with the 
axis, but that the tooth outline twists. The 
appearance of the tooth on either rim, as 
well as upon any section between the two 
rims, is the same as upon a common bevel 
gear, symmetrical, and not canted to one 
side, as is sometimes taught. 

The approximate tooth is very similar to 
the twisted bevel tooth, see (155) and (99), 
with the twist following a straight line set 
askew with the axis, and as the line of the 
twist is not parallel with the conical face, 
that face should be as short as possible. 

The process of cutting is not capable of 
description, for it depends upon personal 
skill and judgment. The workman must 
imagine that he sees the twisted cut in the 
body of the blank, and then must persuade 



the cutter to follow it. Gear cutting ma- 
chines are seldom so made that the cutter 
can be turned while it feeds, and therelore 
it must be set to a medium path, and reset 
two or three times to get the desired form. 
The beginner will fail the first time, and 
there may be several failures. The best pos- 
sible result can be bettered with a file, after 
running the cut gears together to find where 
they interfere. 

In the hands of a skillful workman, a pass- 
able approximation can be reached, and if the 
axes are very near together compared with the 
diameters of the gears, the teeth are small, and 
the face is short, the result is satisfactory. 
In fact, when the conditions are favorable, 
this approximate tooth is more serviceable 
than the true tooth. 




Substitute train. 

Fig. 165. 



180. — SUBSTITUTES FOR THE SKEW BEVEL 
GEAR. 

When there is a chance to introduce an 
intermediate shaft, the skew bevel gear can 
be avoided, and it is not only better, but 
cheaper to avoid the objectionable gear at 
the cost of the extra mechanism. 

Fig. 164 shows how to place an inter- 
mediate shaft and gears, when the shafts are 
so far apart that the shortest or gorge 
distance can be used. Fig. 165 shows how 
the skew shafts can be connected by one 
pair of bevel gears and one pair of spur 
gears, and that is the best device for general 
purposes. 




Substitute train* 

Fig. 164. 



Skew Pin Gears. 



103 



181. — SKEW PLN GEARING. 



Fig. 166 shows a pair of skew pin gears 
commonly called face gears. They will run 
together with a uniform velocity ratio if 
they are exactly alike and at right angles 
with shafts at a distance apart equal to the 
diameter of the pins. 

If the gears are not alike or not at right 
angles, the teeth on one may be straight 
pins, but those on the other must be shaped 
to correspond. 

Such gears are objectionable because they 
have but a single point of contact for each 
pair of teeth, at which they slide on each 
other with great friction. 

Face gearing in its various forms is thor- 
oughly examined in MacCord's Kinematics. 
At the present day they are not in use, and 
do not deserve much study. 




Skew pin Vevel gears. 

Fig. 166. 




Beale gears corrected for interference. 
Fig. 168. 




Formation of Beale gear. 
Fig. 167. 




Uncorrected 
Beale gears. 

Fig. 169. 




I 



1 



